diff --git a/lectures/_static/quant-econ.bib b/lectures/_static/quant-econ.bib index 3e99001d..eaa1b26a 100644 --- a/lectures/_static/quant-econ.bib +++ b/lectures/_static/quant-econ.bib @@ -3734,3 +3734,141 @@ @article{McKayNakamuraSteinsson2016 number = {10}, pages = {3133--3158} } + +@article{BassettoHuoRiosRull2024, + author = {Bassetto, Marco and Huo, Zhen and R{\'i}os-Rull, Jos{\'e}-V{\'i}ctor}, + title = {Institution Building without Commitment}, + journal = {American Economic Review}, + volume = {114}, + number = {11}, + pages = {3427--3468}, + year = {2024} +} + +@techreport{BassettoHuoRiosRull2026, + author = {Bassetto, Marco and Huo, Zhen and R{\'i}os-Rull, Jos{\'e}-V{\'i}ctor}, + title = {Gradual Liberalization and Resilient Trade Cooperation}, + institution = {Rice University, Yale University, and University of Pennsylvania}, + type = {Working paper}, + year = {2026} +} + +@article{Strotz1956, + author = {Strotz, Robert H.}, + title = {Myopia and Inconsistency in Dynamic Utility Maximization}, + journal = {Review of Economic Studies}, + volume = {23}, + number = {3}, + pages = {165--180}, + year = {1956} +} + +@article{PhelpsPollak1968, + author = {Phelps, Edmund S. and Pollak, Robert A.}, + title = {On Second-Best National Saving and Game-Equilibrium Growth}, + journal = {Review of Economic Studies}, + volume = {35}, + number = {2}, + pages = {185--199}, + year = {1968} +} + +@article{Laibson1997, + author = {Laibson, David}, + title = {Golden Eggs and Hyperbolic Discounting}, + journal = {Quarterly Journal of Economics}, + volume = {112}, + number = {2}, + pages = {443--477}, + year = {1997} +} + +@article{KrusellSmith2003, + author = {Krusell, Per and Smith, Anthony A.}, + title = {Consumption-Savings Decisions with Quasi-Geometric Discounting}, + journal = {Econometrica}, + volume = {71}, + number = {1}, + pages = {365--375}, + year = {2003} +} + +@article{JacksonYariv2014, + author = {Jackson, Matthew O. and Yariv, Leeat}, + title = {Present Bias and Collective Dynamic Choice in the Lab}, + journal = {American Economic Review}, + volume = {104}, + number = {12}, + pages = {4184--4204}, + year = {2014} +} + +@article{Kocherlakota1996, + author = {Kocherlakota, Narayana R.}, + title = {Reconsideration-Proofness: A Refinement for Infinite Horizon Time Inconsistency}, + journal = {Games and Economic Behavior}, + volume = {15}, + number = {1}, + pages = {33--54}, + year = {1996} +} + +@article{DekleEatonKortum2008, + author = {Dekle, Robert and Eaton, Jonathan and Kortum, Samuel}, + title = {Global Rebalancing with Gravity: Measuring the Burden of Adjustment}, + journal = {IMF Staff Papers}, + volume = {55}, + number = {3}, + pages = {511--540}, + year = {2008} +} + +@article{BondPark2002, + author = {Bond, Eric W. and Park, Jee-Hyeong}, + title = {Gradualism in Trade Agreements with Asymmetric Countries}, + journal = {Review of Economic Studies}, + volume = {69}, + number = {2}, + pages = {379--406}, + year = {2002} +} + +@article{Bond2009, + author = {Bond, Eric W.}, + title = {Paths of Efficient Self-Enforcing Trade Agreements}, + journal = {Economic Theory}, + volume = {41}, + number = {1}, + pages = {85--104}, + year = {2009} +} + +@incollection{Mussa1986trade, + author = {Mussa, Michael}, + title = {The Adjustment Process and the Timing of Trade Liberalization}, + booktitle = {Economic Liberalization in Developing Countries}, + editor = {Choksi, Armeane M. and Papageorgiou, Demetris}, + publisher = {Basil Blackwell}, + address = {Oxford}, + year = {1986} +} + +@article{Rustichini1998, + author = {Rustichini, Aldo}, + title = {Dynamic Programming Solution of Incentive Constrained Problems}, + journal = {Journal of Economic Theory}, + volume = {78}, + number = {2}, + pages = {329--354}, + year = {1998} +} + +@article{ThomasWorrall1994, + author = {Thomas, Jonathan and Worrall, Tim}, + title = {Foreign Direct Investment and the Risk of Expropriation}, + journal = {Review of Economic Studies}, + volume = {61}, + number = {1}, + pages = {81--108}, + year = {1994} +} diff --git a/lectures/_toc.yml b/lectures/_toc.yml index 2af8758c..3c916a06 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -94,6 +94,8 @@ parts: - file: atkeson_1991 - file: tsyrennikov_2013 - file: dovis_accounting_mf + - file: gradual_trade_agreements + - file: organizational_equilibrium - caption: Other numbered: true chapters: diff --git a/lectures/gradual_trade_agreements.md b/lectures/gradual_trade_agreements.md new file mode 100644 index 00000000..d8f0eaaf --- /dev/null +++ b/lectures/gradual_trade_agreements.md @@ -0,0 +1,973 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst + format_version: 0.13 + jupytext_version: 1.17.1 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + +(gradual_trade_agreements)= +```{raw} jupyter +
+ + QuantEcon + +
+``` + +# Gradualism in Self-Enforcing Trade Agreements + +```{contents} Contents +:depth: 2 +``` + +## Overview + +Free-trade agreements rarely eliminate tariffs at once. + +The North American Free Trade Agreement phased out tariffs between the United States and Mexico over +15 years, the United Kingdom's accession to the European Community eliminated internal tariffs over +five years, and GATT negotiations lowered tariffs round by round. + +If governments could commit to tariffs and if goods and factor markets worked well, immediate free trade +would be efficient {cite}`Mussa1986trade`. + +Governments cannot commit, so trade agreements must be **self-enforcing**: each government must prefer +to honor the agreement, given that a deviation would trigger a trade war. + +This lecture studies two papers that use this requirement to explain gradualism. + +* {cite:t}`BondPark2002` show that when two countries differ in size, the efficient self-enforcing + agreement can start with a positive tariff and move toward free trade, even though the economic + environment never changes. +* {cite:t}`Bond2009` shows that self-enforcement can make it efficient to add members to an agreement + one at a time. + +Bond and Park compute the frontier of self-enforcing agreements recursively, with the payoff promised to +one country as the state variable. + +That makes their problem one of **dynamic programming squared**: the value function of the problem +appears inside its own constraints, as in {doc}`chang_credible`, {doc}`atkeson_1991`, and +{doc}`tsyrennikov_2013`. + +The key force is **backloading**: an agreement relaxes the incentive constraint of the country tempted to +deviate by promising it more in the future, and a tariff imposed by its partner, which the tempted country +cannot escape by deviating, serves as a transfer in the meantime. + +The next lecture, {doc}`organizational_equilibrium`, studies gradual liberalization with a different +equilibrium concept, one that does not rely on the threat of a permanent trade war. + +We use the following imports. + +```{code-cell} ipython3 +import numpy as np +import matplotlib.pyplot as plt +from collections import namedtuple +from scipy.optimize import minimize_scalar +``` + +(gta_model)= +## A two-country trade model + +There are two goods and two countries, a small country $S$ and a large country $L$. + +In country $S$, demand and supply of good $i$ are + +$$ +D_i^S(p) = A - B p , \qquad X_i^S(p) = \alpha_i^S + \beta p , +$$ + +and in country $L$ they are $\lambda$ times as large, + +$$ +D_i^L(p) = \lambda (A - B p) , \qquad X_i^L(p) = \lambda (\alpha_i^L + \beta p ) , +$$ + +where $\lambda \ge 1$ measures the relative size of $L$. + +Following {cite:t}`BondPark2002`, we set $\alpha_1^L = \alpha_2^S > \alpha_2^L = \alpha_1^S$, so that $S$ +imports good 1 and exports good 2, and the countries are mirror images when $\lambda = 1$. + +Each country taxes its imports with a specific tariff: $t_S$ on good 1 imported by $S$, and $t_L$ on good +2 imported by $L$. + +Domestic prices satisfy $p_1^S = p_1^L + t_S$ and $p_2^L = p_2^S + t_L$, and world market clearing for +each good gives + +$$ +p_1^L = \frac{(1 + \lambda) A - \alpha_1^S - \lambda \alpha_1^L - (B + \beta) t_S}{(1 + \lambda)(B + \beta)} , +\qquad +p_2^S = \frac{(1 + \lambda) A - \alpha_2^S - \lambda \alpha_2^L - \lambda (B + \beta) t_L}{(1 + \lambda)(B + \beta)} . +$$ + +National welfare $W_j(t_j, t_k)$ is the sum of consumer surplus, producer surplus, and tariff revenue over +the two goods. + +{cite:t}`BondPark2002` show that + +* $W_j$ is concave in the country's own tariff and is maximized at + + $$ + t_L^N = \frac{\lambda (\alpha_2^S - \alpha_2^L)}{(1 + 2\lambda)(\beta + B)} , \qquad + t_S^N = \frac{\alpha_1^L - \alpha_1^S}{(2 + \lambda)(\beta + B)} ; + $$ + +* $W_j$ is decreasing and convex in the other country's tariff; +* world welfare $W_S + W_L$ is maximized at free trade. + +Because the welfare of each country is additively separable in the two tariffs, $t_j^N$ is a dominant +strategy, and the pair $(t_S^N, t_L^N)$ is the Nash equilibrium of the one-shot tariff game. + +The game is a prisoner's dilemma: each country gains by taxing imports, which improves its terms of trade, +but both would be better off at free trade, at least when they are not too different in size. + +The large country's Nash tariff rises with $\lambda$, and the small country's falls toward zero, because a +small country cannot move world prices. + +```{code-cell} ipython3 +TradeModel = namedtuple('TradeModel', ('A', 'B', 'β', 'α_hi', 'α_lo', 'λ', 'δ')) + +def create_trade_model(A=100.0, B=1.0, β=1.0, α_hi=-40.0, α_lo=-60.0, λ=1.1, δ=0.8): + "α_hi = α_1^L = α_2^S and α_lo = α_2^L = α_1^S." + return TradeModel(A=A, B=B, β=β, α_hi=α_hi, α_lo=α_lo, λ=λ, δ=δ) + +def welfare(t_S, t_L, m): + "National welfare (W_S, W_L) when S taxes good 1 at t_S and L taxes good 2 at t_L." + A, B, β, α_hi, α_lo, λ = m.A, m.B, m.β, m.α_hi, m.α_lo, m.λ + p_1L = ((1 + λ) * A - α_lo - λ * α_hi - (B + β) * t_S) / ((1 + λ) * (B + β)) + p_2S = ((1 + λ) * A - α_hi - λ * α_lo - λ * (B + β) * t_L) / ((1 + λ) * (B + β)) + p_1S, p_2L = p_1L + t_S, p_2S + t_L + CS = lambda p: (A - B * p)**2 / (2 * B) # consumer surplus + PS = lambda p, α: (α + β * p)**2 / (2 * β) # producer surplus + imports = lambda p, α: (A - B * p) - (α + β * p) + W_S = (CS(p_1S) + PS(p_1S, α_lo) + CS(p_2S) + PS(p_2S, α_hi) + + t_S * imports(p_1S, α_lo)) + W_L = λ * (CS(p_1L) + PS(p_1L, α_hi) + CS(p_2L) + PS(p_2L, α_lo) + + t_L * imports(p_2L, α_lo)) + return W_S, W_L + +def nash_tariffs(m): + A, B, β, α_hi, α_lo, λ = m.A, m.B, m.β, m.α_hi, m.α_lo, m.λ + t_LN = λ * (α_hi - α_lo) / ((1 + 2 * λ) * (β + B)) + t_SN = (α_hi - α_lo) / ((2 + λ) * (β + B)) + return t_SN, t_LN +``` + +We check the formulas for the Nash tariffs by maximizing each country's welfare numerically. + +```{code-cell} ipython3 +m = create_trade_model() +t_SN, t_LN = nash_tariffs(m) +best_S = minimize_scalar(lambda t: -welfare(t, 0, m)[0], bounds=(0, 20), method='bounded').x +best_L = minimize_scalar(lambda t: -welfare(0, t, m)[1], bounds=(0, 20), method='bounded').x +print(f"Nash tariffs: t_S = {t_SN:.3f} (numerical {best_S:.3f}), t_L = {t_LN:.3f} " + f"(numerical {best_L:.3f})") +print("welfare at free trade:", np.round(welfare(0, 0, m), 1), + " at the Nash tariffs:", np.round(welfare(t_SN, t_LN, m), 1)) +``` + +(gta_self_enforcing)= +## Self-enforcing agreements + +A **trade agreement** is a sequence of tariffs $t_j(i) \ge 0$ and side payments $s_j(i) \ge 0$ that +country $j$ pays at date $i = 0, 1, \ldots$. + +Side payments stand for concessions in side agreements, such as rules on intellectual property or +foreign investment. + +The payoff to country $j$ from date $i$ on is + +$$ +V_j(i) = \sum_{n=0}^\infty \delta^n \left[ W_j\big(t_j(i+n), t_k(i+n)\big) + s_k(i+n) - s_j(i+n) \right] . +$$ + +If governments could commit, efficient agreements would eliminate tariffs immediately and use side +payments to divide the gains, so the **first-best frontier** is + +$$ +\Omega^*(V_L) = \frac{W_S(0, 0) + W_L(0, 0)}{1 - \delta} - V_L . +$$ + +Without commitment, an agreement must be self-enforcing. + +Following {cite:t}`BondPark2002`, we assume that a deviation triggers permanent reversion to the Nash +tariffs without side payments, which gives country $j$ the payoff $V_j^N = W_j(t_j^N, t_k^N) / (1 - +\delta)$. + +A deviating country sets its Nash tariff and stops paying side payments, but it still receives the side +payment of its partner in the period of deviation. + +The **incentive constraint** of country $j$ at each date $i$ is therefore + +```{math} +:label: gta-ic +V_j(i) \ge W_j\big(t_j^N, t_k(i)\big) + s_k(i) + \delta V_j^N . +``` + +### Stationary free-trade agreements + +Consider free trade with constant side payments. + +The incentive constraint {eq}`gta-ic` becomes $s_j - \delta s_k \le H_j$, where + +$$ +H_j = \delta \left[ W_j(0, 0) - W_j(t_j^N, t_k^N) \right] - (1 - \delta) \left[ W_j(t_j^N, 0) - W_j(0, 0) \right] +$$ + +weighs the future loss from a trade war against the one-period gain from deviating. + +Since only one country pays in an efficient agreement, the largest net side payment that country $j$ is +willing to make each period is $T_j = H_j$ if $H_j > 0$ and $T_j = H_j / \delta$ if $H_j \le 0$, in which +case $j$ must be paid at least $-T_j$. + +A free-trade agreement with constant side payments is sustainable if and only if $T_S + T_L \ge 0$, and +then it can give $L$ any payoff in the interval $[V_L^*, V_L^{**}]$, where + +$$ +V_L^* = \frac{W_L(0, 0) - T_L}{1 - \delta} , \qquad V_L^{**} = \frac{W_L(0, 0) + T_S}{1 - \delta} . +$$ + +Below $V_L^*$, the large country gets too little to stay in a free-trade agreement; above $V_L^{**}$, the +small country does. + +```{code-cell} ipython3 +def stationary_bounds(m): + "Nash payoffs, maximal side payments, and the interval [V_L*, V_L**] of sustainable free trade." + δ = m.δ + t_SN, t_LN = nash_tariffs(m) + W_S0, W_L0 = welfare(0, 0, m) + W_SN, W_LN = welfare(t_SN, t_LN, m) + H_S = δ * (W_S0 - W_SN) - (1 - δ) * (welfare(t_SN, 0, m)[0] - W_S0) + H_L = δ * (W_L0 - W_LN) - (1 - δ) * (welfare(0, t_LN, m)[1] - W_L0) + T_S = H_S if H_S > 0 else H_S / δ + T_L = H_L if H_L > 0 else H_L / δ + return dict(V_SN=W_SN / (1 - δ), V_LN=W_LN / (1 - δ), T_S=T_S, T_L=T_L, + V_L_low=(W_L0 - T_L) / (1 - δ), V_L_high=(W_L0 + T_S) / (1 - δ)) + +models = {"λ = 1.1, δ = 0.8": create_trade_model(λ=1.1, δ=0.8), + "λ = 1.5, δ = 0.7": create_trade_model(λ=1.5, δ=0.7)} +for name, m_i in models.items(): + sb = stationary_bounds(m_i) + print(f"{name}: T_S = {sb['T_S']:.2f}, T_L = {sb['T_L']:.2f}; " + f"V_L* - V_L^N = {sb['V_L_low'] - sb['V_LN']:.2f}, " + f"V_L** - V_L^N = {sb['V_L_high'] - sb['V_LN']:.2f}") +``` + +We study two cases.[^fig1a] + +When $\lambda = 1.1$ and $\delta = 0.8$, both $T_S$ and $T_L$ are positive: each country would pay +something to keep free trade. + +When $\lambda = 1.5$ and $\delta = 0.7$, $T_L < 0$, so the small country must pay for access to the +large country's market; the large country also wins the tariff war, in the sense that it does better at +the Nash equilibrium than at free trade. + +(gta_recursive)= +## A recursive formulation + +Outside $[V_L^*, V_L^{**}]$, stationary free-trade agreements are not self-enforcing. + +To find the efficient agreements there, {cite:t}`BondPark2002` treat the payoff promised to the large +country as a state variable. + +Let $\tilde \Omega(V_L)$ be the largest payoff that the small country can get from a self-enforcing +agreement that gives the large country $V_L$. + +It satisfies + +```{math} +:label: gta-bellman +\tilde \Omega(V_L) = \max_{t_L, t_S, s_L, s_S, y} \; W_S(t_S, t_L) + s_L - s_S + \delta \tilde \Omega(y) +``` + +subject to + +$$ +\begin{aligned} +W_L(t_L, t_S) + s_S - s_L + \delta y &\ge V_L , \\ +W_L(t_L, t_S) - W_L(t_L^N, t_S) - s_L + \delta y &\ge \delta V_L^N , \\ +W_S(t_S, t_L) - W_S(t_S^N, t_L) - s_S + \delta \tilde \Omega(y) &\ge \delta V_S^N , +\end{aligned} +$$ + +where $y$ is the payoff promised to $L$ from next period on. + +The first constraint delivers the promised payoff to $L$, and the other two are the incentive constraints +{eq}`gta-ic` of $L$ and $S$. + +The unknown function $\tilde \Omega$ appears in the objective and in the small country's incentive +constraint, which is what makes this dynamic programming squared. + +When the promise-keeping constraint binds, the net side payment is $s_S - s_L = V_L - W_L - \delta y$, and +the objective becomes + +$$ +W_S(t_S, t_L) + W_L(t_L, t_S) - V_L + \delta \left[ y + \tilde \Omega(y) \right] , +$$ + +world welfare today plus the discounted world value of the continuation, minus what is promised to $L$. + +{cite:t}`BondPark2002` show that $\tilde \Omega$ is the limit of the sequence obtained by applying the +right side of {eq}`gta-bellman` repeatedly to the first-best frontier, building on {cite:t}`Rustichini1998`. + +They also show that a country never imposes a tariff while making a side payment, and their +characterization implies that at most one country imposes a tariff at a time, so we search over one tariff +at a time. + +```{code-cell} ipython3 +def solve_frontier(m, n_V=601, n_t=161, tol=1e-9, max_iter=100): + """ + Iterate on the Bellman equation for the incentive-constrained frontier, starting + from the first-best frontier, on a grid of promised payoffs to L. + """ + δ = m.δ + t_SN, t_LN = nash_tariffs(m) + sb = stationary_bounds(m) + W_S0, W_L0 = welfare(0, 0, m) + V_L_max = welfare(0, t_LN, m)[1] - W_L0 + sb['V_L_high'] # see the text + V = np.unique(np.r_[np.linspace(sb['V_LN'], V_L_max, n_V), sb['V_L_low'], sb['V_L_high']]) + # among equally good continuations, prefer free trade in [V_L*, V_L**] + tie_break = 1e-7 * ((V >= sb['V_L_low']) & (V <= sb['V_L_high'])) + + # candidate tariff pairs: one tariff at a time + t_grid = np.linspace(0, 1, n_t) + t_S = np.r_[np.zeros(n_t), t_SN * t_grid[1:]] + t_L = np.r_[t_LN * t_grid, np.zeros(n_t - 1)] + W_S, W_L = welfare(t_S, t_L, m) + gain_L = welfare(0, t_L, m)[1] - welfare(0, t_LN, m)[1] # W_L(t_L, ·) - W_L(t_L^N, ·) + gain_S = welfare(t_S, 0, m)[0] - welfare(t_SN, 0, m)[0] + + Ω = (W_S0 + W_L0) / (1 - δ) - V # first-best frontier + for it in range(max_iter): + Ω_new = np.empty_like(Ω) + best = np.empty((len(V), 2), dtype=int) + for j, v in enumerate(V): + net = v - W_L[:, None] - δ * V[None, :] # s_S - s_L, promise kept + s_S, s_L = np.maximum(net, 0), np.maximum(-net, 0) + feasible = ((gain_L[:, None] - s_L + δ * V[None, :] >= δ * sb['V_LN'] - 1e-9) + & (gain_S[:, None] - s_S + δ * Ω[None, :] >= δ * sb['V_SN'] - 1e-9)) + obj = np.where(feasible, (W_S + W_L)[:, None] - v + δ * (V + Ω)[None, :], -np.inf) + k = np.argmax(obj + tie_break[None, :]) + best[j] = np.unravel_index(k, obj.shape) + Ω_new[j] = obj[tuple(best[j])] + # L may also be given more than promised: take the best over payoffs V' >= V + source = np.empty(len(V), dtype=int) + source[-1] = len(V) - 1 + for j in range(len(V) - 2, -1, -1): + source[j] = j if Ω_new[j] >= Ω_new[source[j + 1]] else source[j + 1] + Ω_new = Ω_new[source] + change = np.max(np.abs(Ω_new - Ω)) + Ω = Ω_new + if change < tol: + break + i_t, i_y = best[source, 0], best[source, 1] + return dict(V=V, Ω=Ω, t_S=t_S[i_t], t_L=t_L[i_t], y=V[i_y], iterations=it + 1, **sb) + +solutions = {name: solve_frontier(m_i) for name, m_i in models.items()} +for name, sol in solutions.items(): + print(f"{name}: converged after {sol['iterations']} iterations") +``` + +The iteration converges after one application of the Bellman operator, followed by a check that a +second application changes nothing. + +The reason is the main result of {cite:t}`BondPark2002`: the continuation $y$ always lies where +first-best payoffs are attainable, so the continuation value is the first-best one. + +Several continuations are then equally good, and the code picks one in $[V_L^*, V_L^{**}]$, where free +trade with constant side payments is self-enforcing. + +For payoffs above $V_L^{**}$, where only the small country's incentive constraint binds, +{cite:t}`BondPark2002` also give the frontier in closed form, which we use to check the computation. + +Along it, the small country's incentive constraint binds, so $\tilde \Omega = W_S(t_S^N, t_L) + \delta V_S^N$, +and the large country gets $V_L = W_L(t_L, 0) - W_L(0, 0) + V_L^{**}$, for $t_L \in [0, t_L^N]$; the +largest payoff to $L$, used to set the grid above, is the one with $t_L = t_L^N$. + +```{code-cell} ipython3 +for name, m_i in models.items(): + sol = solutions[name] + t_SN, t_LN = nash_tariffs(m_i) + t_check = np.linspace(0, t_LN, 9) + V_L = welfare(0, t_check, m_i)[1] - welfare(0, 0, m_i)[1] + sol['V_L_high'] + Ω_closed = welfare(t_SN, t_check, m_i)[0] + m_i.δ * sol['V_SN'] + gap = np.abs(np.interp(V_L, sol['V'], sol['Ω']) - Ω_closed).max() + print(f"{name}: max |computed - closed form| above V_L** = {gap:.3f}") +``` + +The small differences come from the grids of tariffs and promised payoffs. + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Frontiers of trade agreements + name: fig-gta-frontier +--- +fig, axes = plt.subplots(2, 2, figsize=(12, 8)) +for col, (name, sol) in enumerate(solutions.items()): + V, Ω = sol['V'], sol['Ω'] + W0 = sum(welfare(0, 0, models[name])) + ax = axes[0, col] + ax.plot(V - sol['V_LN'], W0 / (1 - models[name].δ) - V - sol['V_SN'], 'k--', lw=1, + label="first best") + ax.plot(V - sol['V_LN'], Ω - sol['V_SN'], 'C0', lw=2, label="self-enforcing") + ax.set_title(name) + ax.set_xlabel(r"$V_L - V_L^N$") + ax.set_ylabel(r"$V_S - V_S^N$") + ax = axes[1, col] + ax.plot(V - sol['V_LN'], sol['t_S'], 'C1', lw=2, label=r"first-period $t_S$") + ax.plot(V - sol['V_LN'], sol['t_L'], 'C2', lw=2, label=r"first-period $t_L$") + ax.set_xlabel(r"$V_L - V_L^N$") + ax.set_ylabel("tariff") + for a in axes[:, col]: + for x in [sol['V_L_low'], sol['V_L_high']]: + a.axvline(x - sol['V_LN'], color='gray', ls=':', lw=1) +axes[0, 0].legend() +axes[1, 0].legend() +plt.tight_layout() +plt.show() +``` + +The top panels show the two frontiers, measured from the Nash payoffs; the dotted lines mark $V_L^*$ and +$V_L^{**}$. + +Between them, free trade with constant side payments is self-enforcing and the two frontiers coincide. + +Above $V_L^{**}$, the small country is asked to pay so much that it would rather deviate. + +The efficient agreement then lets the large country tax its imports in the first period: the small +country can stop a side payment by deviating, but it cannot escape a tariff. + +From the second period on, the agreement switches to free trade with a payoff to $L$ in +$[V_L^*, V_L^{**}]$, so the tariff falls in two steps, from $t_L^N$ to a first-period tariff and then to +zero. + +Below $V_L^*$, the same logic applies with the roles reversed, and the small country imposes the +first-period tariff. + +When $T_L < 0$ (right panels), there is a range below $V_L^*$ in which first-best payoffs can still be +reached without tariffs, by lowering the small country's first-period side payment. + +The constrained frontier is strictly below the first-best frontier wherever a tariff is used, and in +every case the payoff of the constrained country *rises* over time: the agreement promises it more later +in exchange for less now. + +(gta_smoothing)= +## Consumption smoothing and many-step liberalization + +In the model so far, liberalization takes only two steps. + +{cite:t}`BondPark2002` show that it becomes gradual when the small country wants to smooth its +consumption, for example because it lacks access to international capital markets. + +Let $S$ be infinitesimally small, so that it cannot affect world prices, its Nash tariff is zero, and +$t_S = 0$ in any efficient agreement. + +Prices in $S$ are then $p_1^S = p_1^{LA}$ and $p_2^S = p_2^{LA} - t_L$, where $p_i^{LA} = (A - +\alpha_i^L)/(B + \beta)$ are the large country's autarky prices. + +The large country's welfare from the agreement is its tariff revenue, $W_L(t_L) = t_L \big[ X_2^S(p_2^S) - +D_2^S(p_2^S) \big]$, which is maximized at $t_L^N = (\alpha_2^S - \alpha_2^L) / \big(2 (B + \beta)\big)$. + +The small country's payoff in a period is $U\big(W_S(t_L) + s_L - s_S\big)$, with $U$ strictly concave, +while the large country's payoff stays linear. + +### Where the large country's incentive constraint binds + +With free trade, the large country's payoff is the side payment it receives, and its incentive +constraint requires that it receive at least $W_L(t_L^N) / \delta$ per period, so stationary free trade +is self-enforcing only for $V_L \ge V_L^* = V_L^N / \delta$, where $V_L^N = W_L(t_L^N) / (1 - \delta)$. + +{cite:t}`BondPark2002` show that for $V_L \in (V_L^N, V_L^*)$ the large country's incentive constraint +binds, the small country's does not, and the small country pays a side payment, so $s_L = 0$. + +The binding incentive constraint of $L$ and the promise-keeping constraint then pin down + +$$ +W_L(t_L) + \delta y = V_L^N , \qquad s_S = V_L - V_L^N , +$$ + +so that the only choice left is the tariff, and the Bellman equation becomes + +```{math} +:label: gta-smoothing-bellman +\tilde \Omega(V_L) = \max_{0 \le t_L \le t_L^N} \; U\big( W_S(t_L) - (V_L - V_L^N) \big) ++ \delta \tilde \Omega\left( \frac{V_L^N - W_L(t_L)}{\delta} \right) , +``` + +where $\tilde \Omega(y) = U\big(W_S(0) - (1 - \delta) y\big) / (1 - \delta)$ on the first-best region +$y \in [V_L^*, V_L^{**}]$. + +Because $W_L(t_L) \ge 0$, the continuation $y$ never exceeds $V_L^*$, and it reaches $V_L^*$ only when +$t_L = 0$. + +A tariff $t_L > 0$ lowers the payoff $y$ promised to $L$ for the future, so it slows the rise in the small +country's side payment. + +The agreement trades a static distortion, the tariff, against an intertemporal one, an uneven consumption +path for $S$, and it reduces both gradually. + +We use the demand and supply parameters of the previous section, $\delta = 0.9$, and constant relative +risk aversion, $U(c) = \left[ (c/\bar c)^{1 - \gamma} - 1 \right] / (1 - \gamma)$, with consumption scaled +by its free-trade level $\bar c = W_S(0)$. + +The side payments are small relative to the small country's surplus, less than 4 percent of it, so the +consumption-smoothing motive matters only when curvature is high, and we show results for +$\gamma \in \{5, 20, 50\}$. + +We start each agreement at $V_L = V_L^N$, the point on the frontier that gives the small country all the +gains, and check that the small country's incentive constraint is slack along the solution, as +Proposition 3 of {cite:t}`BondPark2002` asserts. + +```{code-cell} ipython3 +SmallCountry = namedtuple('SmallCountry', ('A', 'B', 'β', 'α_hi', 'α_lo', 'δ', 'γ')) + +def create_small_country(A=100.0, B=1.0, β=1.0, α_hi=-40.0, α_lo=-60.0, δ=0.9, γ=20.0): + return SmallCountry(A=A, B=B, β=β, α_hi=α_hi, α_lo=α_lo, δ=δ, γ=γ) + +def small_country_welfare(t_L, sc): + "The small country's surplus W_S(t_L) and the large country's tariff revenue W_L(t_L)." + A, B, β, α_hi, α_lo = sc.A, sc.B, sc.β, sc.α_hi, sc.α_lo + p_1 = (A - α_hi) / (B + β) # L's autarky price of good 1 + p_2 = (A - α_lo) / (B + β) - t_L # price of good 2 in S + W_S = ((A - B * p_1)**2 / (2 * B) + (α_lo + β * p_1)**2 / (2 * β) + + (A - B * p_2)**2 / (2 * B) + (α_hi + β * p_2)**2 / (2 * β)) + W_L = t_L * ((α_hi + β * p_2) - (A - B * p_2)) + return W_S, W_L + +def solve_smoothing(sc, n_V=600, n_t=600, tol=1e-13, max_iter=5000): + "Iterate on the Bellman equation where L's incentive constraint binds." + δ, γ = sc.δ, sc.γ + t_N = (sc.α_hi - sc.α_lo) / (2 * (sc.B + sc.β)) + W_S0 = small_country_welfare(0.0, sc)[0] + if γ == 1: + U = lambda c: np.log(c / W_S0) + else: + U = lambda c: ((c / W_S0)**(1 - γ) - 1) / (1 - γ) + V_N = small_country_welfare(t_N, sc)[1] / (1 - δ) + V_low = V_N / δ + Ω_first_best = lambda y: U(W_S0 - (1 - δ) * y) / (1 - δ) + + V = np.linspace(V_N, V_low, n_V) + t = np.linspace(0, t_N, n_t) + W_S, W_L = small_country_welfare(t, sc) + y = (V_N - W_L) / δ # L's binding incentive constraint + Ω = Ω_first_best(V) + for it in range(max_iter): + Ωy = np.where(y >= V_low, Ω_first_best(y), np.interp(y, V, Ω)) + obj = U(W_S[None, :] - (V[:, None] - V_N)) + δ * Ωy[None, :] + Ω_new = obj.max(axis=1) + if np.max(np.abs(Ω_new - Ω)) < tol: + Ω = Ω_new + break + Ω = Ω_new + + def policy(v): + "The optimal tariff at promised payoff v, by continuous maximization." + def rhs(tt): + W_S_t, W_L_t = small_country_welfare(tt, sc) + yy = (V_N - W_L_t) / δ + Ωy = Ω_first_best(yy) if yy >= V_low else np.interp(yy, V, Ω) + return U(W_S_t - (v - V_N)) + δ * Ωy + return minimize_scalar(lambda tt: -rhs(tt), bounds=(0, t_N), method='bounded', + options={'xatol': 1e-10}).x + + # slack in the small country's incentive constraint along the grid solution + k = obj.argmax(axis=1) + Ωy_k = np.where(y[k] >= V_low, Ω_first_best(y[k]), np.interp(y[k], V, Ω)) + V_SN = U(small_country_welfare(t_N, sc)[0]) / (1 - δ) + ics_slack = (U(W_S[k] - (V - V_N)) + δ * Ωy_k) - (U(W_S[k]) + δ * V_SN) + + return dict(V=V, Ω=Ω, policy=policy, V_N=V_N, V_low=V_low, t_N=t_N, iterations=it + 1, + min_ics_slack=ics_slack.min()) + +def simulate_agreement(sc, sol, n=30): + "Tariffs, side payments, and promised payoffs, starting from V_L = V_L^N." + v, path = sol['V_N'], [] + for i in range(n): + if v >= sol['V_low']: # free trade from here on + path.append((0.0, v - sol['V_N'], v)) + continue + t_L = sol['policy'](v) + path.append((t_L, v - sol['V_N'], v)) + v = (sol['V_N'] - small_country_welfare(t_L, sc)[1]) / sc.δ + return np.array(path) + +γ_values = [5, 20, 50] +paths = {} +for γ in γ_values: + sc = create_small_country(γ=γ) + sol = solve_smoothing(sc) + paths[γ] = simulate_agreement(sc, sol) + print(f"γ = {γ}: converged after {sol['iterations']} iterations; smallest slack in S's " + f"incentive constraint {sol['min_ics_slack']:.3f}; first tariffs {np.round(paths[γ][:5, 0], 3)}") +``` + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Gradual liberalization with consumption smoothing + name: fig-gta-smoothing +--- +sc = create_small_country() +sol_ref = solve_smoothing(sc) +W_S_free = small_country_welfare(0.0, sc)[0] +fig, axes = plt.subplots(1, 3, figsize=(14, 4)) +for γ in γ_values: + t_L, s_S = paths[γ][:, 0], paths[γ][:, 1] + consumption = small_country_welfare(t_L, sc)[0] - s_S + axes[0].plot(t_L, 'o-', ms=3, lw=2, label=rf"$\gamma = {γ}$") + axes[1].plot(s_S, 'o-', ms=3, lw=2) + axes[2].plot(100 * (consumption / W_S_free - 1), 'o-', ms=3, lw=2) +axes[1].axhline(sol_ref['V_low'] - sol_ref['V_N'], color='gray', ls=':', lw=1) +for ax, title in zip(axes, ["large country's tariff $t_L$", + r"side payment $s_S = V_L - V_L^N$", + "small country's consumption (% from free trade)"]): + ax.set_title(title) + ax.set_xlabel("period") +axes[0].legend() +plt.tight_layout() +plt.show() +``` + +The agreement starts with a positive tariff by the large country that declines period by period toward +zero. + +The side payment from the small country, which equals $V_L - V_L^N$, rises toward $V_L^* - V_L^N$, the +dotted line, so the payoff promised to the large country rises toward the region where free trade is +self-enforcing. + +The small country's consumption, its surplus net of the side payment, falls gradually rather than all at +once. + +The more the small country cares about smoothing its consumption, the higher the first tariff and the +slower the liberalization. + +Without curvature, the large country would be compensated entirely by a side payment that jumps after +the first period, and no tariff would be used in this region; curvature makes a jump in side payments +costly, so the agreement uses a declining tariff to slow their rise. + +(gta_membership)= +## Gradual expansion of membership + +{cite:t}`Bond2009` asks a related question about who belongs to an agreement: when is it efficient to +form a bilateral agreement first and to add a third country later? + +There are three countries and three states: no agreement $N$, a bilateral agreement $B$ between +countries 1 and 2, and a global agreement $G$. + +A country gets the flow payoff $w(N)$, $w(G)$, or, under $B$, $w_m(B)$ as a member and $w_n(B)$ as the +non-member, and it pays a one-time adjustment cost $k$ when it joins an agreement, for example the cost +of reallocating workers and of adapting its laws. + +### Commitment + +If countries could commit to agreements and side payments, an efficient path would maximize world +welfare, the sum of the countries' payoffs net of adjustment costs. + +Let $w^W(a)$ be the sum of the countries' flow payoffs in state $a$, so that $w^W(G) = 3 w(G)$ and $w^W(B) = +2 w_m(B) + w_n(B)$, and let $C(a', a)$ be the sum of their adjustment costs of moving from $a$ to $a'$. + +Starting from $N$, forming $B$ at date 0 and $G$ at date 1 beats forming $G$ at once if the adjustment +costs it saves exceed the world welfare it forgoes for one period: + +$$ +C(G, N) - C(B, N) - \delta C(G, B) > w^W(G) - w^W(B) . +$$ + +{cite:t}`Bond2009` assumes that a permanent move to $G$ beats staying in any other state $a$ forever, +$w^W(G) - (1 - \delta) C(G, a) > w^W(a)$. + +When each country pays the same cost $k$ whenever it joins an agreement, the left side of the inequality +above is $(1 - \delta) k$, and the assumption for $a = B$ says that $(1 - \delta) k < w^W(G) - w^W(B)$, so +forming $G$ at once maximizes world welfare. + +Efficient delay with commitment needs adjustment costs that depend more strongly on the sequence of +agreements, for example when adding a country costs less once a bilateral agreement is in place. + +### Self-enforcement + +Without commitment, each member of an agreement must prefer to honor it. + +A member that deviates from an agreement $a$ gets $w^D(a)$ for one period, avoids the adjustment cost of +joining, keeps any side payment it was supposed to pay, still receives any side payment owed to it, and +then gets a punishment payoff $w^P(a)$ forever. + +The **enforcement power** of an agreement for a member, $\Phi(a, \delta)$, is the excess of the member's +average payoff from the agreement over its average payoff from deviating, before side payments and +adjustment costs: $\Phi(G, \delta) = w(G) - \left[ (1 - \delta) w^D(G) + \delta w^P(G) \right]$, and +$\Phi(B, \delta)$ is defined in the same way with the member's flow payoff $w_m(B)$. + +{cite:t}`Bond2009` shows that side payments should be *backloaded*: a side payment received at the date a +country joins does not relax its incentive constraint, because it would receive it even if it deviated, so +compensation should come in later periods. + +A country $i$ that joins with adjustment cost $c_i$ can then be asked to pay at most + +$$ +s_i^{\max} = \min\left\{ \Phi(a, \delta), \; \frac{\Phi(a, \delta) - (1 - \delta) c_i}{\delta} \right\} +$$ + +per period from the next period on, and the agreement is self-enforcing if and only if the maximal +payments add up to a nonnegative number. + +In an **immediate global agreement**, every country pays $k$, so the agreement is self-enforcing if and +only if $\Phi(G, \delta) \ge (1 - \delta) k$. + +A **two-step agreement** forms $B$ at date 0 and adds country 3 at date 1. + +At date 1, countries 1 and 2 are already members, so each can pay up to $\Phi(G, \delta)$ per period from +date 2 on to subsidize the newcomer, whose maximal payment is $\left[ \Phi(G, \delta) - (1 - \delta) k +\right] / \delta$. + +The second step is therefore self-enforcing if and only if $(1 + 2\delta) \Phi(G, \delta) \ge (1 - \delta) +k$.[^bond-two-step] + +The smallest subsidy that each original member must then pay is $s = \max\left\{0, \left[ (1 - \delta) k - +\Phi(G, \delta) \right] / (2\delta) \right\}$, so from date 1 on an original member gets the average payoff +$w(G) - \delta s$. + +At date 0, each of countries 1 and 2 pays $k$ and gets $w_m(B)$, so its incentive constraint is + +```{math} +:label: gta-first-step +\Phi(B, \delta) - (1 - \delta) k + \delta \left[ w(G) - \delta s - w_m(B) \right] \ge 0 . +``` + +The original members can subsidize the newcomer out of the slack enforcement power of the global +agreement, which makes the second step easier to sustain than an immediate global agreement. + +They must also be willing to stay in the first step, which is easier when the bilateral agreement has +more enforcement power than the global one. + +We compute the smallest discount factors that sustain each path for an example in which the bilateral +agreement has more enforcement power, $\Phi(B, \delta) = \Phi(G, \delta) + 1$, because a member of the +bilateral agreement gains less by deviating and is punished more harshly. + +```{code-cell} ipython3 +Membership = namedtuple('Membership', ('w_G', 'w_D_G', 'w_P_G', 'w_mB', 'w_nB', 'w_D_B', 'w_P_B', 'k')) + +def create_membership(w_G=12.0, w_D_G=14.0, w_P_G=10.0, w_mB=12.0, w_nB=9.5, + w_D_B=13.0, w_P_B=9.0, k=4.0): + return Membership(w_G=w_G, w_D_G=w_D_G, w_P_G=w_P_G, w_mB=w_mB, w_nB=w_nB, + w_D_B=w_D_B, w_P_B=w_P_B, k=k) + +def immediate_slack(δ, mb): + Φ_G = mb.w_G - ((1 - δ) * mb.w_D_G + δ * mb.w_P_G) + return Φ_G - (1 - δ) * mb.k + +def two_step_slack(δ, mb): + "The smaller of the slacks in the second-step and first-step constraints." + Φ_G = mb.w_G - ((1 - δ) * mb.w_D_G + δ * mb.w_P_G) + Φ_B = mb.w_mB - ((1 - δ) * mb.w_D_B + δ * mb.w_P_B) + second = (1 + 2 * δ) * Φ_G - (1 - δ) * mb.k + s = max(0.0, ((1 - δ) * mb.k - Φ_G) / (2 * δ)) # subsidy per original member + first = Φ_B - (1 - δ) * mb.k + δ * (mb.w_G - δ * s - mb.w_mB) + return min(second, first, Φ_G) + +def min_discount_factor(slack, mb): + "The smallest δ on a fine grid at which the slack is nonnegative." + grid = np.linspace(0.01, 0.999, 9_000) + ok = np.array([slack(δ, mb) >= 0 for δ in grid]) + return grid[ok.argmax()] if ok.any() else np.nan + +mb = create_membership() +print(f"smallest δ: immediate global agreement {min_discount_factor(immediate_slack, mb):.3f}, " + f"two-step agreement {min_discount_factor(two_step_slack, mb):.3f}") +δ = 0.7 +print(f"at δ = {δ}: average world welfare of the two-step path minus the immediate path = " + f"{(1 - δ) * ((1 - δ) * mb.k - (3 * mb.w_G - 2 * mb.w_mB - mb.w_nB)):.3f}") +``` + +In this example, the global agreement can be reached in two steps for discount factors at which an +immediate global agreement would not be self-enforcing, even though, with commitment, the immediate path +would yield higher world welfare. + +The bilateral agreement is a **building bloc**: for discount factors between the two thresholds, without +it the global agreement would never be reached. + +The advantage comes from the bilateral agreement's extra enforcement power: {ref}`gta_ex3` shows that +with the same enforcement power as the global agreement, the two-step path would need the same discount +factor as the immediate one. + +{cite:t}`Bond2009` shows more generally that whether the two-step path maximizes world welfare is neither +necessary nor sufficient for it to be easier to sustain, because sustainability depends on deviation +payoffs as well as on world welfare. + +(gta_connections)= +## Connections to other lectures + +* {doc}`organizational_equilibrium` also explains gradual trade liberalization, but with an equilibrium + concept that does not rely on the threat of a permanent trade war; here, as in the best plans sustained + by Markov reversion in that lecture, a single deviation ends cooperation forever. +* {doc}`calvo_abreu` and {doc}`chang_credible` study sustainable plans; {cite:t}`BondPark2002` note that + replacing Nash reversion with the harshest credible punishments of {cite:t}`Abreu` would enlarge the set + of self-enforcing agreements without changing their shape over time. +* {doc}`atkeson_1991` and {doc}`tsyrennikov_2013` study lending with moral hazard and a risk of + repudiation, in which a value function enters the enforcement constraint; the backloading of payments to + the party whose incentive constraint binds is the force that {cite:t}`ThomasWorrall1994` found in + contracts with a host country that cannot commit not to expropriate. +* {doc}`repeat_mh` and {doc}`un_insure` use promised utility as the state variable in problems with hidden + actions instead of limited commitment. + +## Exercises + +```{exercise} +:label: gta_ex1 + +The relative size $\lambda$ of the large country determines who pays whom in a self-enforcing +free-trade agreement. + +Holding $\delta = 0.8$ and the other parameters fixed, compute $T_S$, $T_L$, and the large country's gain +from the tariff war, $W_L(t_L^N, t_S^N) - W_L(0, 0)$, for $\lambda$ between 1 and 2. + +1. For which $\lambda$ does the large country win the tariff war? +1. For which $\lambda$ can a free-trade agreement have the large country pay the small one? +``` + +```{solution-start} gta_ex1 +:class: dropdown +``` + +```{code-cell} ipython3 +λ_grid = np.linspace(1, 2, 101) +T_S, T_L, war_gain = [], [], [] +for λ_i in λ_grid: + m_i = create_trade_model(λ=λ_i, δ=0.8) + sb = stationary_bounds(m_i) + T_S.append(sb['T_S']) + T_L.append(sb['T_L']) + war_gain.append(welfare(*nash_tariffs(m_i), m_i)[1] - welfare(0, 0, m_i)[1]) +T_S, T_L, war_gain = map(np.array, (T_S, T_L, war_gain)) + +fig, ax = plt.subplots() +ax.plot(λ_grid, T_S, lw=2, label=r"$T_S$") +ax.plot(λ_grid, T_L, lw=2, label=r"$T_L$") +ax.plot(λ_grid, war_gain, lw=2, label=r"$W_L^N - W_L(0, 0)$") +ax.axhline(0, color='k', lw=0.8) +ax.set_xlabel(r"relative size $\lambda$") +ax.legend() +plt.show() + +print(f"L wins the tariff war for λ above {λ_grid[np.argmax(war_gain > 0)]:.2f}") +print(f"T_L > 0 for λ below {λ_grid[np.argmax(T_L < 0)]:.2f}") +``` + +The large country wins the tariff war once it is about 38 percent larger than the small country, and +from then on any free-trade agreement must have the small country pay for access. + +The large country is willing to pay for free trade only when the countries are close in size, for +$\lambda$ below about 1.2 when $\delta = 0.8$; a larger $\lambda$ raises its gain from deviating relative to +the cost of the trade war. + +At $\delta = 0.8$, the small country is willing to pay throughout this range, and more so the larger its +partner. + +```{solution-end} +``` + +```{exercise} +:label: gta_ex2 + +In {ref}`gta_smoothing`, compute the number of periods it takes for the large country's tariff to fall +below one percent of its first-period value, for $\gamma \in \{5, 20, 50\}$ and $\delta \in \{0.85, 0.9, +0.95\}$. + +How do the speed of liberalization and the first-period tariff depend on $\gamma$ and $\delta$? +``` + +```{solution-start} gta_ex2 +:class: dropdown +``` + +```{code-cell} ipython3 +print(" δ γ first tariff periods to fall below 1% of it") +for δ_i in [0.85, 0.9, 0.95]: + for γ in [5, 20, 50]: + sc = create_small_country(δ=δ_i, γ=γ) + path = simulate_agreement(sc, solve_smoothing(sc), n=200) + t_path = path[:, 0] + n_below = np.argmax(t_path < 0.01 * t_path[0]) + print(f"{δ_i:5.2f} {γ:4d} {t_path[0]:11.3f} {n_below:12d}") +``` + +More curvature raises the first-period tariff and slows liberalization, because smoothing the small +country's consumption matters more. + +A higher discount factor lowers the first-period tariff and slightly speeds liberalization. + +```{solution-end} +``` + +```{exercise} +:label: gta_ex3 + +In {ref}`gta_membership`, vary the adjustment cost $k$ from 0 to 8 and compute + +1. the smallest discount factors that sustain the immediate and the two-step paths, and +1. at $\delta = 0.8$, whether the two-step path would yield higher world welfare under commitment. + +Find values of $k$ at which the two-step path is easier to sustain even though it yields lower world +welfare, and repeat the calculation with $w^D(B) = 14$ and $w^P(B) = 10$, which give the bilateral +agreement the same enforcement power as the global one. +``` + +```{solution-start} gta_ex3 +:class: dropdown +``` + +```{code-cell} ipython3 +def step_slacks(δ, mb): + "Slacks in the second-step and first-step constraints of the two-step path." + Φ_G = mb.w_G - ((1 - δ) * mb.w_D_G + δ * mb.w_P_G) + Φ_B = mb.w_mB - ((1 - δ) * mb.w_D_B + δ * mb.w_P_B) + s = max(0.0, ((1 - δ) * mb.k - Φ_G) / (2 * δ)) + return (1 + 2 * δ) * Φ_G - (1 - δ) * mb.k, Φ_B - (1 - δ) * mb.k + δ * (mb.w_G - δ * s - mb.w_mB) + +δ = 0.8 +print(" k smallest δ: immediate two-step (binding step) same enforcement power" + " two-step better with commitment") +for k in [0, 1, 2, 4, 6, 8]: + mb_k = create_membership(k=k) + mb_same = create_membership(k=k, w_D_B=14.0, w_P_B=10.0) + δ_two = min_discount_factor(two_step_slack, mb_k) + second, first = step_slacks(δ_two, mb_k) + binding = "first" if first < second else "second" + better = (1 - δ) * k > 3 * mb_k.w_G - 2 * mb_k.w_mB - mb_k.w_nB + print(f"{k:4.0f} {min_discount_factor(immediate_slack, mb_k):19.3f} {δ_two:8.3f}" + f" {'(' + binding + ')':<8s} {min_discount_factor(two_step_slack, mb_same):18.3f}" + f" {str(better):>14s}") +``` + +With these payoffs, the two-step path never yields higher world welfare at $\delta = 0.8$: postponing the +newcomer's adjustment cost saves $(1 - \delta) k$, less than the 2.5 that the bilateral period loses, which +is the assumption of {cite:t}`Bond2009` discussed in {ref}`gta_membership` and holds for $k < 12.5$. + +Yet for every positive adjustment cost, the two-step path is self-enforcing at lower discount factors +than the immediate path. + +For small $k$, the second step sets the threshold, so the subsidy to the newcomer limits how far below +the immediate threshold the two-step path can go; for $k \ge 4$, the first step sets it. + +In every case, the gain requires slack in the first step, which here comes from the bilateral agreement's +extra enforcement power: when the bilateral agreement has the same enforcement power as the global one, +the two-step path needs exactly the same discount factor as the immediate path, whatever $k$. + +```{solution-end} +``` + +[^bond-two-step]: {cite:t}`Bond2009` states this condition as $3 \Phi(G, \delta) = (1 - \delta) k$; + applying his Lemma 1 to the newcomer, whose side payment at the entry date does not relax its + constraint, gives the factor $1 + 2\delta$ used here. His statement of the first-step condition + coincides with {eq}`gta-first-step` when the newcomer needs a subsidy, $s > 0$. + +[^fig1a]: The caption of Figure 1(a) of {cite:t}`BondPark2002` reports $\lambda = 1.1$, $\delta = 0.65$, and + $T_L > 0$; with these parameters we find $T_L \approx -1.01$, and $T_L$ becomes positive only for + $\delta$ above about 0.7, so we use $\delta = 0.8$. diff --git a/lectures/organizational_equilibrium.md b/lectures/organizational_equilibrium.md new file mode 100644 index 00000000..e34079ce --- /dev/null +++ b/lectures/organizational_equilibrium.md @@ -0,0 +1,1191 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst + format_version: 0.13 + jupytext_version: 1.17.1 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + +(organizational_equilibrium)= +```{raw} jupyter +
+ + QuantEcon + +
+``` + +# Institution Building Without Commitment + +```{contents} Contents +:depth: 2 +``` + +## Overview + +Many policies that experts regard as good, such as lower tariffs and taxes on carbon emissions, have been +adopted slowly, in steps spread over decades. + +{cite:t}`BassettoHuoRiosRull2024` attribute the slowness to a problem of time inconsistency. + +A fast reform imposes its costs on the current policymaker and delivers its benefits to later ones, so +each policymaker would like its successors to carry out the reform. + +Several lectures in this section study this kind of tension with three equilibrium concepts. + +* A **Ramsey planner** chooses a whole sequence of actions once and for all at time $0$, as in + {doc}`calvo`, {doc}`dyn_stack`, and {doc}`opt_tax_recur`. +* A sequence of **Markov** policymakers each choose a current action, taking as given that future + actions do not depend on it, as in the Markov perfect governments of {doc}`calvo`. +* A sequence of policymakers follow a **sustainable plan**, in which expectations of how future + policymakers will react to a deviation deter deviations, as in {doc}`calvo_abreu` and + {doc}`chang_credible`. + +{cite:t}`BassettoHuoRiosRull2024` propose a fourth equilibrium concept, **organizational equilibrium**. + +It refines the set of subgame-perfect equilibria of the game among successive policymakers, the game +behind sustainable plans, with two requirements designed to deny period $0$ the special role that it has in +a Ramsey plan. + +* *No restarting*: no policymaker can do better by pretending to be an earlier policymaker and + restarting the plan. +* *No delay*: the first policymaker cannot do better by sitting out, choosing its best one-period action, + and letting the plan start one period later. + +Among the equilibria that satisfy these requirements, an organizational equilibrium is the best one. + +An organizational equilibrium has three striking features. + +* Policy improves *gradually*, as if good institutions had to earn trust over time. +* Policy converges to the best constant policy, which lies between the Markov and Ramsey policies. +* No permanent reversion to a bad equilibrium is needed to support it: after an unexpected deviation, + policy can resume its gradual improvement from where the deviation left it. + +{cite:t}`BassettoHuoRiosRull2026` use the last feature to explain why trade liberalization has survived +recurrent but temporary episodes of protection. + +This lecture + +* computes the Ramsey, Markov, and best constant policies in a growth model with present-biased + preferences, the motivating example of {cite:t}`BassettoHuoRiosRull2024`, +* defines an organizational equilibrium and computes it in that model, +* computes organizational equilibrium tariffs in a two-country model of trade and growth, and compares + them with Markov tariffs, Ramsey tariffs, and tariffs sustained by the threat of reverting to the Markov + equilibrium, including after a temporary protectionist shock, and +* describes connections to other lectures in this section. + +We use the following imports. + +```{code-cell} ipython3 +import numpy as np +import matplotlib.pyplot as plt +from collections import namedtuple +from scipy.optimize import brentq, minimize, minimize_scalar +``` + +(oe_growth)= +## A savings problem with present bias + +A planner at each date $t$ chooses consumption $c_t$ and next period's capital $k_{t+1}$ subject to + +$$ +c_t + k_{t+1} = k_t^\alpha , \qquad k_0 \text{ given}, +$$ + +with $0 < \alpha < 1$; capital depreciates fully. + +The planner at date $t$ ranks consumption sequences by + +$$ +\log c_t + \delta \sum_{j=1}^\infty \beta^j \log c_{t+j} , +$$ + +with $0 < \beta < 1$ and $0 < \delta \le 1$. + +When $\delta < 1$, preferences are **quasi-geometric**, or present-biased: the planner at date $t$ discounts +the next period by $\delta \beta$ but each later period by only $\beta$ +{cite}`Strotz1956,PhelpsPollak1968,Laibson1997,KrusellSmith2003`. + +The planner at date $t+1$ discounts date $t+2$ more heavily relative to date $t+1$ than the planner at +date $t$ wants it to, so the two planners disagree, and the date-$t$ plan is time inconsistent. + +{cite:t}`BassettoHuoRiosRull2024` interpret the present bias as the outcome of aggregating +the preferences of current and future generations, or of parties with different discount factors, as in +{cite:t}`JacksonYariv2014`. + +### Separating the state from the actions + +Let $s_t = k_{t+1} / k_t^\alpha$ be the saving rate, so that $c_t = (1 - s_t) k_t^\alpha$. + +A sequence of saving rates and $k_0$ determine $\log k_t = \alpha^t \log k_0 + \sum_{j=0}^{t-1} +\alpha^{t-1-j} \log s_j$. + +Substituting into the planner's objective shows that the payoff of the planner at date $t$ is the sum of a +term that depends only on capital and a term that depends only on current and future saving rates: + +```{math} +:label: oe-separable +U(k_t, s_t, s_{t+1}, \ldots) = \frac{\alpha (1 - \alpha\beta + \delta\alpha\beta)}{1 - \alpha\beta} +\log k_t + V(s_t, s_{t+1}, \ldots) , +``` + +where the **action payoff** is + +```{math} +:label: oe-action-payoff +V(s_t, s_{t+1}, \ldots) = \log(1 - s_t) + \frac{\delta\alpha\beta}{1 - \alpha\beta} \log s_t ++ \delta \sum_{j=1}^\infty \beta^j \left[ \log(1 - s_{t+j}) + \frac{\alpha\beta}{1 - \alpha\beta} +\log s_{t+j} \right] . +``` + +This separability is the key to the analysis. + +It implies that every planner ranks sequences of saving rates in the same way whatever capital it +inherits, and that every sequence of saving rates is feasible from every level of capital. + +So we can compare what different planners get from their own and their successors' actions, even when +they inherit different capital stocks, by comparing action payoffs. + +The next function computes the action payoff of a sequence of saving rates that becomes constant after a +finite number of periods. + +```{code-cell} ipython3 +α, β, δ = 0.4, 0.96, 0.9 # the parameter values of Bassetto, Huo, and Ríos-Rull + +def action_payoff(s, s_tail, α=α, β=β, δ=δ): + """ + The action payoff V(s_0, s_1, ...) of the saving rates in the list s, + followed by s_tail forever. + """ + a = α * β / (1 - α * β) + u = lambda x: np.log(1 - x) + a * np.log(x) + W = u(s_tail) / (1 - β) # value of the constant tail + for x in s[:0:-1]: # s_{n-1}, ..., s_1 + W = u(x) + β * W + return np.log(1 - s[0]) + δ * a * np.log(s[0]) + δ * β * W +``` + +### Three benchmarks + +A Ramsey planner at date $0$ chooses the whole sequence of saving rates once and for all. + +It saves at the rate + +$$ +s_0^R = \frac{\alpha\delta\beta}{1 - \alpha\beta + \delta\alpha\beta} +$$ + +at date $0$ and at the rate $s^R = \alpha\beta$ at every later date. + +Because it discounts date $1$ more heavily than later dates, the date-$0$ planner saves less itself than it +wants its successors to save. + +In a Markov equilibrium, each planner chooses its own saving rate, taking as given that its successors' +saving rates do not depend on its choice. + +Separability makes the best response of each planner independent of what its successors do, so the +Markov equilibrium has the constant saving rate + +```{math} +:label: oe-sM +s^M = \frac{\alpha\delta\beta}{1 - \alpha\beta + \delta\alpha\beta} , +``` + +which equals the Ramsey planner's saving rate at date $0$. + +A third benchmark is the **best constant saving rate**, the rate that a planner would choose if it had +to choose one rate for itself and all its successors: + +```{math} +:label: oe-sB +s^B = \arg\max_s V(s, s, s, \ldots) += \frac{\delta\alpha\beta}{(1 - \beta + \delta\beta)(1 - \alpha\beta) + \delta\alpha\beta} . +``` + +When $0 < \delta < 1$, $s^M < s^B < s^R$. + +We compute the three rates and check two claims numerically: the formula for $s^B$, and the claim that +the Markov best response does not depend on what successors do. + +```{code-cell} ipython3 +def benchmarks(α=α, β=β, δ=δ): + "Saving rates in the Markov equilibrium, at the best constant rate, and in the Ramsey plan." + s_M = α * δ * β / (1 - α * β + δ * α * β) + s_B = δ * α * β / ((1 - β + δ * β) * (1 - α * β) + δ * α * β) + return s_M, s_B, α * β + +s_M, s_B, s_R = benchmarks() +print(f"s_M = {s_M:.4f}, s_B = {s_B:.4f}, s_R = {s_R:.4f}") + +bounds = (1e-6, 1 - 1e-6) +best = minimize_scalar(lambda x: -action_payoff([x], x), bounds=bounds, method='bounded', + options={'xatol': 1e-12}).x +print(f"numerical maximizer of V(s, s, ...) = {best:.6f}") +for s_future in [0.1, 0.3, 0.6]: + br = minimize_scalar(lambda x: -action_payoff([x], s_future), bounds=bounds, + method='bounded', options={'xatol': 1e-12}).x + print(f"best response when successors save {s_future}: {br:.6f}") +``` + +(oe_definition)= +## Organizational equilibrium + +### Three properties + +Consider the three benchmarks from the point of view of planners who can see what earlier planners got. + +In the Ramsey plan, the date-$0$ planner gets the action payoff $V(s^M, s^R, s^R, \ldots)$, while every +later planner gets $V(s^R, s^R, \ldots)$, which is smaller. + +Any later planner would rather be the date-$0$ planner, if it could restart the plan. + +The same is true of the best subgame-perfect equilibrium sustained by the threat of reverting to the +Markov equilibrium after a deviation, which also treats the date-$0$ planner better than its successors. + +The best constant saving rate treats all planners alike, but it tempts the date-$0$ planner to sit out: +$V(s^M, s^B, s^B, \ldots) > V(s^B, s^B, \ldots)$, so the date-$0$ planner would rather save at its +Markov rate and let the plan start next period. + +The Markov equilibrium survives both tests: it treats all planners alike, and a planner who sits out +chooses $s^M$ and gets exactly its equilibrium payoff. + +An organizational equilibrium is a sequence of saving rates $\{s_t^\ast\}$ with three properties. + +* **No restarting**: every planner gets the same action payoff, + + ```{math} + :label: oe-no-restart + V(s_t^\ast, s_{t+1}^\ast, \ldots) = \bar V \quad \text{for all } t . + ``` + +* **Optimality**: no other sequence that satisfies {eq}`oe-no-restart` attains a higher $\bar V$. +* **No delay**: the first planner does not gain by choosing its best one-period action and letting the + sequence start one period later, + + ```{math} + :label: oe-no-delay + V(s_0^\ast, s_1^\ast, \ldots) \ge \max_s V(s, s_0^\ast, s_1^\ast, \ldots) . + ``` + +Why must the action payoff be constant? + +A planner at date $t+1$ that could become the planner at date $t$ would do so if +$V(s_t^\ast, s_{t+1}^\ast, \ldots) > V(s_{t+1}^\ast, s_{t+2}^\ast, \ldots)$, so action payoffs must be +weakly increasing over time; if they ever rose, the first planner would do better with the sequence that +starts where they are higher, contradicting optimality. + +### Game-theoretic foundations + +{cite:t}`BassettoHuoRiosRull2024` define an organizational equilibrium as the outcome of a +subgame-perfect equilibrium of the game among the sequence of planners that satisfies three requirements: + +1. strategies do not depend on the physical state $k$; +1. every planner gets the same action payoff after every history, and no other equilibrium with this + property gives a higher payoff; +1. the first planner has no incentive to sit out and let the equilibrium start one period later. + +Without a state variable, the second requirement is the **reconsideration-proofness** of +{cite:t}`Kocherlakota1996`; separability is what lets Bassetto, Huo, and Ríos-Rull extend it to economies +with capital. + +They prove that a sequence with the three properties of the previous subsection is an organizational +equilibrium, and, under further conditions that hold in the growth model, that + +* its action payoff is the highest that a constant action can attain; +* its payoff lies strictly between those of the best Markov equilibrium and the Ramsey plan whenever the + Ramsey plan is not itself a Markov equilibrium; +* the continuation value of the sequence rises over time and converges to that of the best constant + action, but not immediately. + +The no-delay requirement can look like an arbitrary refinement. + +They show that it is instead a *necessary* property of equilibrium in an alternative game in which no one +knows when record keeping started and any planner can hide the past and pretend to be the first. + +In that game, a planner who faced a sequence that violated no delay would pass the buck: it would play +its one-shot best response and let its successor believe that the plan was just starting. + +The equilibrium can be supported by a strategy that restarts the sequence after a deviation, but a milder +punishment suffices: after a planner deviates to the action that an earlier planner took, the +continuation can resume the sequence from that action. + +### Constant-value paths + +Subtracting $\beta V(s_{t+1}, s_{t+2}, \ldots)$ from {eq}`oe-action-payoff` and imposing +{eq}`oe-no-restart` gives a first-order difference equation for the saving rate: + +```{math} +:label: oe-q +\beta (1 - \delta) \log(1 - s_{t+1}) += \frac{\delta\alpha\beta}{1 - \alpha\beta} \log s_t + \log(1 - s_t) - (1 - \beta) \bar V , +``` + +which we write as $s_{t+1} = q(s_t; \bar V)$. + +Many sequences deliver the same action payoff, because a planner who saves more today is compensated by +more saving by its successors. + +{cite:t}`BassettoHuoRiosRull2024` show that the highest constant value that any sequence can attain is +$V^\ast = V(s^B, s^B, \ldots)$, the value of the best constant saving rate. + +With $\bar V = V^\ast$, the map $q$ has $s^B$ as a fixed point, and paths that start between $s^M$ and $s^B$ +rise toward it. + +```{code-cell} ipython3 +V_star = action_payoff([s_B], s_B) + +def q(s, V_bar, α=α, β=β, δ=δ): + "Next period's saving rate on a path whose action payoff is V_bar at every date." + a = α * β / (1 - α * β) + return 1 - np.exp((-(1 - β) * V_bar + δ * a * np.log(s) + np.log(1 - s)) / (β * (1 - δ))) + +s_0 = q(s_M, V_star) +slope = lambda s: (q(s + 1e-7, V_star) - q(s - 1e-7, V_star)) / 2e-7 +print(f"s_0 = q(s_M; V*) = {s_0:.4f}") +print(f"slope of q at s_M: {abs(slope(s_M)):.4f}, at s_B: {slope(s_B):.4f}") +``` + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Saving rates on constant-payoff paths + name: fig-oe-q +--- +grid = np.linspace(s_M - 0.005, s_B + 0.002, 300) +fig, ax = plt.subplots() +ax.plot(grid, q(grid, V_star), 'C3', lw=2, label=r"$q(s; V^*)$") +for gap in [0.003, 0.006]: + ax.plot(grid, q(grid, V_star - gap), 'C0--', lw=1) +ax.plot(grid, grid, 'k', lw=0.8, label="45-degree line") +for x, name in [(s_M, r"$s^M$"), (s_0, r"$s_0$"), (s_B, r"$s^B$")]: + ax.axvline(x, color='gray', ls=':', lw=1) + ax.text(x, ax.get_ylim()[0], name, ha='center', va='bottom') +ax.set_xlabel(r"$s_t$") +ax.set_ylabel(r"$s_{t+1}$") +ax.legend() +plt.show() +``` + +The red curve is $q(s; V^\ast)$, and the dashed curves are $q(s; \bar V)$ for two values of $\bar V$ below +$V^\ast$. + +The red curve is flat at $s^M$ and touches the 45-degree line at $s^B$ with slope one, so convergence slows +as the saving rate approaches $s^B$. + +### Where the path starts + +The difference equation {eq}`oe-q` admits a continuum of solutions with value $V^\ast$, one for each +starting point. + +All of them give the first planner the same payoff, but they differ in what they ask of it. + +A sequence that starts at $s_0 = s^B$ is constant and, as we saw, violates no delay. + +The first planner's best one-period action is $s^M$, whatever its successors do, so the no-delay condition +{eq}`oe-no-delay` requires the starting point to be low enough that saving at $s^M$ and letting the sequence +start next period does not pay. + +The highest starting point that passes is $s_0 = q(s^M; V^\ast)$: with it, the sequence +$s^M, s_0, s_1, \ldots$ is itself a constant-value path, so sitting out gives exactly $V^\ast$. + +Bassetto, Huo, and Ríos-Rull select this starting point because it leaves the most capital to later +planners. + +The organizational equilibrium therefore starts with a saving rate about halfway from the Markov rate to +$s^B$ and raises it gradually toward $s^B$, which is below the Ramsey planner's long-run rate $s^R$ for any +$\delta < 1$. + +### Transition paths + +We compare the organizational equilibrium with the Ramsey plan and the Markov equilibrium, starting from +the Markov steady-state capital stock $k^M = (s^M)^{1/(1 - \alpha)}$, as +{cite:t}`BassettoHuoRiosRull2024` do. + +The **total payoff** of the planner at date $t$ is {eq}`oe-separable`, the sum of a capital payoff and +the action payoff $V_t$. + +Along the organizational path, $V_t = V^\ast$ at every date by construction; we confirm it on a long +computed path, since the saving rate converges slowly. + +```{code-cell} ipython3 +n_long, n_plot = 5000, 16 +s_O = [s_0] +for t in range(n_long): + s_O.append(q(s_O[-1], V_star)) +s_O = np.array(s_O) + +V_O = np.array([action_payoff(s_O[t:], s_B) for t in range(n_plot)]) +print(f"max |V_t - V*| over {n_plot} periods = {np.abs(V_O - V_star).max():.1e}") + +paths = {"organizational equilibrium": s_O[:n_plot], + "Ramsey plan": np.r_[s_M, np.full(n_plot - 1, s_R)], + "Markov equilibrium": np.full(n_plot, s_M)} +V_paths = {"organizational equilibrium": V_O, + "Ramsey plan": np.r_[action_payoff([s_M], s_R), + np.full(n_plot - 1, action_payoff([s_R], s_R))], + "Markov equilibrium": np.full(n_plot, action_payoff([s_M], s_M))} +c_k = α * (1 - α * β + δ * α * β) / (1 - α * β) + +def capital_path(s, k_0): + k = [k_0] + for x in s[:-1]: + k.append(x * k[-1]**α) + return np.array(k) + +k_M = s_M**(1 / (1 - α)) +print("V_t - V* under the Ramsey plan, dates 0 and 1:", + np.round(V_paths["Ramsey plan"][:2] - V_star, 6)) +print(f"V_t - V* in the Markov equilibrium: {V_paths['Markov equilibrium'][0] - V_star:.4f}") +``` + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Transition paths in three equilibria + name: fig-oe-growth-paths +--- +fig, axes = plt.subplots(2, 2, figsize=(11, 7)) +styles = {"organizational equilibrium": 'C3-', "Ramsey plan": 'C0--', + "Markov equilibrium": 'k:'} +for name, s in paths.items(): + k = capital_path(s, k_M) + axes[0, 0].plot(s, styles[name], lw=2, label=name) + axes[0, 1].plot(k, styles[name], lw=2) + axes[1, 0].plot(c_k * np.log(k) + V_paths[name], styles[name], lw=2) + if name != "Markov equilibrium": + axes[1, 1].plot(V_paths[name] - V_star, styles[name], lw=2) +for ax, title in zip(axes.flat, ["saving rate $s_t$", "capital $k_t$", + "total payoff $U_t$", r"action payoff $V_t - V^*$"]): + ax.set_title(title) + ax.set_xlabel("period") +axes[0, 0].legend() +plt.tight_layout() +plt.show() +``` + +In the Markov equilibrium, the saving rate and capital stay at their initial levels. + +The Ramsey planner saves at the Markov rate at date $0$ and then imposes the higher rate $s^R$ on all its +successors. + +In the organizational equilibrium, the date-$0$ planner saves more than the Markov rate, so capital at +date $1$ is higher than under the Ramsey plan, and the saving rate then rises gradually. + +The Ramsey plan gives the date-$0$ planner the highest total payoff, as it must. + +At date $1$ the comparison reverses: the date-$1$ planner inherits more capital and is asked to save +less than under the Ramsey plan, so its total payoff is higher. + +From date $2$ on, capital is lower than under the Ramsey plan, and so are total payoffs. + +The action payoffs, which factor out inherited capital, show the redistribution (the Markov equilibrium's +action payoff, $0.044$ below $V^\ast$, is left out of the last panel): an organizational equilibrium gives +the first planner less than the Ramsey plan does and gives every later planner more. + +(oe_tariffs)= +## Tariffs, saving, and growth + +We apply these ideas to trade policy, following {cite:t}`BassettoHuoRiosRull2024` and +{cite:t}`BassettoHuoRiosRull2026`. + +In this application a single large player, the government, interacts with a continuum of competitive +private agents. + +Bassetto, Huo, and Ríos-Rull handle this setting by assuming that every sequence of government actions +induces a unique competitive equilibrium that does not depend on the initial state, and by defining the +government's preferences over sequences of its own actions as its payoff in that competitive equilibrium; +the definitions of the previous section then apply unchanged. + +Time inconsistency now has a familiar source: private agents' current decisions depend on expected future +policy, as in {cite:t}`KydlandPrescott1977`, {doc}`calvo`, and {doc}`dyn_stack`. + +A tariff protects workers in the import-competing sector and raises revenue that is rebated to workers, +but it lowers the return to capital, and thereby saving and growth. + +A policymaker who cares about workers therefore wants high tariffs now and low tariffs later: low future +tariffs encourage saving today, while today's tariff cannot affect saving that has already been done. + +### A two-country economy + +There are two symmetric countries, home and foreign, and two goods. + +Home is more productive at producing good 1, foreign at producing good 2. + +In each country, a measure one of workers is attached to each sector, and capital moves freely between +the two sectors. + +At home, sector $i$ produces + +$$ +Y_{it} = A_i \, k_t^{1 - \alpha} \ell_{it}^{1 - \alpha} k_{it}^\alpha , \qquad \ell_{it} = 1, \quad +A_1 = A > 1, \quad A_2 = 1 , +$$ + +where $\ell_{it}$ is labor and $k_{it}$ capital used in sector $i$, and the country's aggregate capital +stock $k_t = k_{1t} + k_{2t}$ raises productivity in both sectors, an externality that makes output +proportional to $k_t$, as in an $AK$ model. + +Foreign is the mirror image of home, and the two countries start with the same capital stock. + +Each government taxes imports at the rate $\tau_t \ge 0$, and, following {cite:t}`BassettoHuoRiosRull2026`, +we assume that the foreign government matches the home tariff, so that home effectively chooses a common +tariff. + +Let good 2 be the numeraire at home. + +Home imports good 2, so its world price is $1/(1 + \tau_t)$, and by symmetry the world and home prices of +good 1, which home exports, are also $p_{1t} = 1/(1 + \tau_t)$. + +The tariff lowers the relative price of home's export good below its free-trade value of one. + +#### Static equilibrium + +Equating the returns to capital in the two sectors gives the share $\phi_t$ of capital in sector 2, + +$$ +\phi_t = \left[ 1 + \left( \frac{A}{1 + \tau_t} \right)^{1/(1 - \alpha)} \right]^{-1} , +$$ + +so that a higher tariff shifts capital into the import-competing sector. + +Wages and the rental rate, in units of good 2, are + +$$ +w_{1t} = (1 - \alpha) \frac{A (1 - \phi_t)^\alpha}{1 + \tau_t} k_t , \qquad +w_{2t} = (1 - \alpha) \phi_t^\alpha k_t , \qquad +r_t = \alpha \phi_t^{\alpha - 1} . +$$ + +A final good used for consumption and investment combines the two goods with a CES aggregator with +equal weights and elasticity of substitution $\varepsilon$, and its price is + +$$ +P_t = \left[ \tfrac12 p_{1t}^{1 - \varepsilon} + \tfrac12 \right]^{1/(1 - \varepsilon)} . +$$ + +Home uses the goods in the ratio $m_{1t} / m_{2t} = (1 + \tau_t)^{\varepsilon}$. + +Because world prices of the two goods are equal, balanced trade requires home's exports of good 1, +$Y_{1t} - m_{1t}$, to equal its imports of good 2, $m_{2t} - Y_{2t}$. + +Home collects tariff revenue $T_t = \tau_t (m_{2t} - Y_{2t}) / (1 + \tau_t)$, which it rebates equally to the +two groups of workers. + +Workers do not save, so a worker in sector $i$ consumes + +$$ +c_{it} = \frac{w_{it} + T_t / 2}{P_t} = \psi_i(\tau_t) \, k_t , +$$ + +where $\psi_i(\tau)$ depends only on the tariff. + +#### Capitalists + +Capitalists own the capital stock and rent it to firms; their consumption and investment satisfy + +$$ +c^K_t + k_{t+1} = G(\tau_t) \, k_t , \qquad G(\tau) = \frac{r(\tau)}{P(\tau)} + 1 - \delta_k , +$$ + +where $\delta_k$ is the depreciation rate, and they maximize $\sum_t \beta^t (c^K_t)^{1 - \sigma} / (1 - +\sigma)$. + +With saving rate $s_t = k_{t+1} / (G(\tau_t) k_t)$, the Euler equation is + +```{math} +:label: oe-euler +\left( \frac{1 - s_t}{s_t} \right)^{-\sigma} = \beta \, G(\tau_{t+1})^{1 - \sigma} (1 - s_{t+1})^{-\sigma} . +``` + +Iterating {eq}`oe-euler` forward shows that $s_t$ depends only on *future* tariffs. + +When $\sigma < 1$, a higher expected return raises saving, so expected future tariffs lower saving today. + +When tariffs are constant, $s(\tau) = \left[ \beta G(\tau)^{1 - \sigma} \right]^{1/\sigma}$. + +Within each period, the government sets the tariff first and capitalists then decide how much to save, so +capitalists respond in the same period to any change in expected policy. + +#### The policymaker + +The home policymaker maximizes the welfare of the average worker, +$\sum_{j \ge 0} \beta^j \tfrac12 (\log c_{1,t+j} + \log c_{2,t+j})$. + +Since $\tfrac12 (\log c_{1t} + \log c_{2t}) = \log k_t + \chi(\tau_t)$, with $\chi(\tau) = \tfrac12 [\log +\psi_1(\tau) + \log \psi_2(\tau)]$, and $\log k_{t+1} = \log k_t + \log s_t + \log G(\tau_t)$, the +policymaker's payoff separates into a capital payoff and an action payoff: + +```{math} +:label: oe-tariff-payoff +U_t = \frac{\log k_t}{1 - \beta} + \sum_{j=0}^\infty \beta^j H(\tau_{t+j}, s_{t+j}) , \qquad +H(\tau, s) = \chi(\tau) + \frac{\beta}{1 - \beta} \left[ \log s + \log G(\tau) \right] . +``` + +Two functions of the tariff alone, worker welfare $\chi(\tau)$ and the gross return $G(\tau)$, summarize +everything that the static equilibrium contributes to the dynamic policy problem. + +We use the parameter values of {cite:t}`BassettoHuoRiosRull2024`, with a model period of seven years, +and set $\sigma = 0.5$, the value used by {cite:t}`BassettoHuoRiosRull2026`. + +```{code-cell} ipython3 +TariffModel = namedtuple('TariffModel', ('α', 'β', 'δ_k', 'ε', 'A', 'σ')) + +def create_tariff_model(α=0.33, β=0.7, δ_k=0.65, ε=1.1, A=3.5, σ=0.5): + return TariffModel(α=α, β=β, δ_k=δ_k, ε=ε, A=A, σ=σ) + +def static(τ, model): + """ + Worker welfare χ(τ), the gross return G(τ), and the consumption of the two + groups of workers per unit of capital, ψ_1(τ) and ψ_2(τ). + """ + α, β, δ_k, ε, A, σ = model + ϕ = 1 / (1 + (A / (1 + τ))**(1 / (1 - α))) + p_1 = 1 / (1 + τ) + Y_1, Y_2 = A * (1 - ϕ)**α, ϕ**α + P = (0.5 * p_1**(1 - ε) + 0.5)**(1 / (1 - ε)) + m_2 = (Y_1 + Y_2) / (1 + (1 + τ)**ε) # balanced trade + revenue = τ / (1 + τ) * (m_2 - Y_2) + ψ_1 = ((1 - α) * p_1 * Y_1 + revenue / 2) / P + ψ_2 = ((1 - α) * Y_2 + revenue / 2) / P + χ = 0.5 * (np.log(ψ_1) + np.log(ψ_2)) + G = α * ϕ**(α - 1) / P + 1 - δ_k + return χ, G, ψ_1, ψ_2 + +def χ(τ, model): + return static(τ, model)[0] + +def G(τ, model): + return static(τ, model)[1] + +def H(τ, s, model): + β = model.β + return χ(τ, model) + β / (1 - β) * (np.log(s) + np.log(G(τ, model))) + +def s_const(τ, model): + "The saving rate when the tariff is τ forever." + return (model.β * G(τ, model)**(1 - model.σ))**(1 / model.σ) + +model = create_tariff_model() +``` + +### Markov and best constant tariffs + +From {eq}`oe-euler`, today's saving rate depends only on future tariffs. + +A Markov policymaker, who takes future tariffs as given, therefore takes today's saving rate as given and +chooses + +$$ +\tau^M = \arg\max_{\tau \ge 0} \; \chi(\tau) + \frac{\beta}{1 - \beta} \log G(\tau) . +$$ + +The best constant tariff instead takes into account that a permanently lower tariff raises the saving +rate $s(\tau)$; substituting $s(\tau)$ into $H$ gives, up to a constant, + +$$ +\tau^B = \arg\max_{\tau \ge 0} \; \chi(\tau) + \frac{\beta}{\sigma (1 - \beta)} \log G(\tau) , +$$ + +which puts $1/\sigma$ times as much weight on the return to capital. + +```{code-cell} ipython3 +def argmax(f, lo=0.0, hi=1.0): + return minimize_scalar(lambda τ: -f(τ), bounds=(lo, hi), method='bounded', + options={'xatol': 1e-10}).x + +def markov_tariff(model): + β = model.β + return argmax(lambda τ: χ(τ, model) + β / (1 - β) * np.log(G(τ, model))) + +def best_constant_tariff(model): + β, σ = model.β, model.σ + return argmax(lambda τ: χ(τ, model) + β / (σ * (1 - β)) * np.log(G(τ, model))) + +τ_M, τ_B = markov_tariff(model), best_constant_tariff(model) +H_B = H(τ_B, s_const(τ_B, model), model) +V_markov = H(τ_M, s_const(τ_M, model), model) / (1 - model.β) +V_best = H_B / (1 - model.β) +print(f"Markov tariff {100 * τ_M:.1f} percent, best constant tariff {100 * τ_B:.1f} percent") + +ψ_free, ψ_markov = static(0.0, model)[2:], static(τ_M, model)[2:] +print("consumption at the Markov tariff relative to free trade: export-sector workers " + f"{ψ_markov[0] / ψ_free[0]:.3f}, import-sector workers {ψ_markov[1] / ψ_free[1]:.3f}") +``` + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Worker welfare and capital returns + name: fig-oe-chi-G +--- +τ_grid = np.linspace(0, 0.5, 200) +fig, axes = plt.subplots(1, 2, figsize=(11, 4)) +axes[0].plot(100 * τ_grid, [χ(x, model) - χ(0, model) for x in τ_grid], lw=2) +axes[0].set_title(r"worker welfare $\chi(\tau) - \chi(0)$") +axes[1].plot(100 * τ_grid, [G(x, model) for x in τ_grid], lw=2) +axes[1].set_title(r"gross return to capital $G(\tau)$") +for ax in axes: + ax.axvline(100 * τ_B, color='C2', ls='--', lw=1, label=r"best constant tariff $\tau^B$") + ax.axvline(100 * τ_M, color='C3', ls=':', lw=1.5, label=r"Markov tariff $\tau^M$") + ax.set_xlabel("tariff (percent)") +axes[0].legend() +plt.tight_layout() +plt.show() +``` + +The tariff raises worker welfare over the whole range shown. + +It raises the consumption of the lower-paid workers in the import-competing sector, through their wage +and the rebated revenue, by much more in proportion than it lowers the consumption of workers in the +export sector: at the Markov tariff, by 58 percent against 6 percent. + +It lowers the return to capital throughout. + +The Markov policymaker, who ignores how expected tariffs affect saving, chooses a high tariff; the best +constant tariff is close to free trade. + +### Ramsey plan and plans sustained by Markov reversion + +A Ramsey planner at date $0$ chooses the whole sequence of tariffs, taking into account the Euler +equations {eq}`oe-euler`. + +Saving at date $t$ depends only on tariffs after date $t$, so the date-$0$ tariff affects no saving rate, +and the planner chooses $\tau_0 = \tau^M$; from date $1$ on, it lowers tariffs to encourage saving at +earlier dates. + +We compute the Ramsey plan by optimizing over tariffs for $N$ periods, after which the tariff is constant. + +Given a tariff sequence, we compute saving rates backward from the steady state using {eq}`oe-euler`, and +action payoffs $V_t = \sum_{j \ge 0} \beta^j H(\tau_{t+j}, s_{t+j})$ backward from the constant tail. + +```{code-cell} ipython3 +def saving_rates(τ, model): + "Saving rates for tariffs τ_0, ..., τ_N, with τ_N kept forever." + β, σ = model.β, model.σ + s = np.empty(len(τ)) + s[-1] = s_const(τ[-1], model) + for t in range(len(τ) - 2, -1, -1): + ratio = (β * G(τ[t + 1], model)**(1 - σ))**(-1 / σ) * (1 - s[t + 1]) # (1 - s_t) / s_t + s[t] = 1 / (1 + ratio) + return s + +def action_payoffs(τ, model): + "Action payoffs V_0, ..., V_N for tariffs τ_0, ..., τ_N, with τ_N kept forever." + β = model.β + s = saving_rates(τ, model) + V = np.empty(len(τ)) + V[-1] = H(τ[-1], s[-1], model) / (1 - β) + for t in range(len(τ) - 2, -1, -1): + V[t] = H(τ[t], s[t], model) + β * V[t + 1] + return V + +N = 25 +res = minimize(lambda τ: -action_payoffs(τ, model)[0], np.full(N + 1, 0.1), + bounds=[(0, 1)] * (N + 1), method='L-BFGS-B') +τ_R = res.x +V_R = action_payoffs(τ_R, model) +print("Ramsey tariffs (percent):", np.round(100 * τ_R[:6], 1)) +print(f"action payoffs: V_0 = {V_R[0]:.4f}, V_1 = {V_R[1]:.4f}, V_N = {V_R[-1]:.4f}; " + f"V* = {V_best:.4f}, V_M = {V_markov:.4f}") +print(f"smallest V_t - V_M along the Ramsey plan: {(V_R - V_markov).min():.4f}") +``` + +The Ramsey plan starts at the Markov tariff and reaches free trade within a few periods. + +Free trade is a corner: we restrict tariffs to be nonnegative, as {cite:t}`BassettoHuoRiosRull2026` do, +and without that restriction the Ramsey plan would subsidize imports from period 4 on. + +The plan gives the date-$0$ policymaker a higher action payoff than any later one, so it violates no +restarting. + +A sustainable plan in the sense of {cite:t}`chari1990sustainable`, the subject of {doc}`calvo_abreu`, can be +supported by the threat that any deviation triggers permanent reversion to the Markov equilibrium. + +After a deviation, capitalists expect Markov tariffs from the next period on and save at the Markov rate, +so the best deviation yields exactly the Markov action payoff $V^M$. + +A tariff path can therefore be sustained by Markov reversion if and only if $V_t \ge V^M$ at every date, +and the best such path maximizes $V_0$ subject to these constraints. + +Every Ramsey continuation is worth more than the Markov equilibrium, as the last line printed above shows, +so the constraints are slack and the best path sustained by Markov reversion is the Ramsey plan. + +The best sustainable plan in the sense of {doc}`calvo_abreu`, which uses the worst subgame-perfect +equilibrium as the punishment, is therefore also the Ramsey plan. + +### Organizational equilibrium tariffs + +Along an organizational equilibrium, the action payoff is the same at every date. + +We take that value to be $V^\ast = H(\tau^B, s^B)/(1 - \beta)$, the value of the best constant tariff, as +{cite:t}`BassettoHuoRiosRull2026` do. + +With a competitive private sector, the general results that make $V^\ast$ the highest constant value in the +growth model do not apply directly, so this choice must be checked in each application; here a numerical +search over tariff paths that repeat every two, three, or four periods finds none that gives every +policymaker more than $V^\ast$. + +A constant action payoff requires $H(\tau_t, s_t) = H^B \equiv H(\tau^B, s^B)$ at every date. + +Solving this equation for the saving rate gives the saving rate that a tariff $\tau$ must induce, + +$$ +s^O(\tau) = \exp\left\{ \frac{1 - \beta}{\beta} \left[ H^B - \chi(\tau) \right] - \log G(\tau) \right\} , +$$ + +and the Euler equation {eq}`oe-euler` then links successive tariffs: + +$$ +\left( \frac{1 - s^O(\tau_t)}{s^O(\tau_t)} \right)^{-\sigma} += \beta \, G(\tau_{t+1})^{1 - \sigma} \left[ 1 - s^O(\tau_{t+1}) \right]^{-\sigma} . +$$ + +For a given $\tau_t$, this equation has a root $\tau_{t+1}$ between $\tau^B$ and $\tau_t$ and another at a +much higher tariff; the first defines the transition map $\tau_{t+1} = \mathcal{T}(\tau_t)$, which has the +fixed point $\tau^B$. + +A high tariff today must be followed by a tariff low enough tomorrow to induce the saving rate that keeps +$H(\tau_t, s_t) = H^B$. + +The no-delay condition requires $\tau_0 \ge \mathcal{T}(\tau^M)$: a policymaker who sat out would choose +$\tau^M$, and sitting out pays if the path starts with a lower tariff. + +As in the growth model, we select the lowest tariff that satisfies it, $\tau_0 = \mathcal{T}(\tau^M)$, as +{cite:t}`BassettoHuoRiosRull2026` do; with it, the sequence $\tau^M, \tau_0, \tau_1, \ldots$ is itself a +path of constant action payoff, so sitting out gains nothing. + +```{code-cell} ipython3 +def s_org(τ, H_B, model): + "The saving rate that keeps H(τ, s) equal to H_B." + β = model.β + return np.exp((1 - β) / β * (H_B - χ(τ, model)) - np.log(G(τ, model))) + +def transition(τ, τ_B, H_B, model): + "Next period's tariff on a path whose action payoff is constant at H_B / (1 - β)." + if τ <= τ_B + 1e-8: + return τ_B + β, σ = model.β, model.σ + s = s_org(τ, H_B, model) + lhs = ((1 - s) / s)**(-σ) + f = lambda τ_next: (β * G(τ_next, model)**(1 - σ) + * (1 - s_org(τ_next, H_B, model))**(-σ) - lhs) + return brentq(f, τ_B, τ) # the root between τ_B and τ + +𝒯 = lambda τ: transition(τ, τ_B, H_B, model) + +τ_O = [𝒯(τ_M)] +for t in range(400): + τ_O.append(𝒯(τ_O[-1])) +τ_O = np.array(τ_O) +V_org = action_payoffs(τ_O, model) +print("organizational tariffs (percent):", np.round(100 * τ_O[:6], 1)) +print(f"max |V_t - V*| over the first 25 periods = {np.abs(V_org[:25] - V_best).max():.1e}") + +ratio = ((model.β * G(τ_O[0], model)**(1 - model.σ))**(-1 / model.σ) + * (1 - s_org(τ_O[0], H_B, model))) +s_minus = 1 / (1 + ratio) # saving at date -1 if the path starts one period later +print(f"gain from sitting out at date 0: H(τ_M, s_-1) - H_B = {H(τ_M, s_minus, model) - H_B:.1e}") +print(f"slope of the transition map at τ_M: {(𝒯(τ_M) - 𝒯(τ_M - 1e-6)) / 1e-6:.4f}, " + f"near τ_B: {(𝒯(τ_B + 2e-3) - 𝒯(τ_B + 1e-3)) / 1e-3:.4f}") +print(f"tariff after 20 periods {100 * τ_O[20]:.1f} percent, after 100 periods " + f"{100 * τ_O[100]:.1f} percent") +``` + +The action payoff is constant along the path, and the first policymaker is exactly indifferent between +starting the organizational path and sitting out. + +Like $q$ in the growth model, the transition map is flat at the Markov policy and has slope one at the +best constant policy, so liberalization is fast at first and then very slow: with seven-year periods, the +tariff is still 3.8 percent after 140 years. + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Tariff transition map and tariff paths + name: fig-oe-tariff-paths +--- +n_plot = 20 +fig, axes = plt.subplots(1, 2, figsize=(12, 4.5)) +grid_τ = np.linspace(τ_B, τ_M, 100) +axes[0].plot(100 * grid_τ, [100 * 𝒯(x) for x in grid_τ], 'C3', lw=2, + label=r"$\mathcal{T}(\tau)$") +axes[0].plot(100 * grid_τ, 100 * grid_τ, 'k', lw=0.8, label="45-degree line") +axes[0].set_xlabel(r"current tariff $\tau_t$ (percent)") +axes[0].set_ylabel(r"next tariff $\tau_{t+1}$ (percent)") +axes[0].legend() +axes[1].plot(100 * τ_O[:n_plot], 'C3-o', ms=4, lw=2, label="organizational equilibrium") +axes[1].plot(100 * τ_R[:n_plot], 'C0--s', ms=4, lw=2, + label="Ramsey plan = best plan sustained by Markov reversion") +axes[1].plot(np.full(n_plot, 100 * τ_M), 'k:', lw=2, label="Markov equilibrium") +axes[1].axhline(100 * τ_B, color='gray', lw=0.8) +axes[1].set_xlabel("period (seven years)") +axes[1].set_ylabel("tariff (percent)") +axes[1].legend() +plt.tight_layout() +plt.show() +``` + +The transition map lies below the 45-degree line, so tariffs fall at every date, by a lot when they are +high and by little near $\tau^B$. + +The organizational equilibrium starts well below the Markov tariff and liberalizes gradually, a pattern +that {cite:t}`BassettoHuoRiosRull2024` compare with the decline of the world average tariff after 1950. + +The Ramsey plan liberalizes much faster, but only because its first policymaker can commit its +successors. + +### Temporary protectionist shocks + +The off-path behavior of the two cooperative equilibria differs sharply. + +Suppose that at date 10 an unexpected shock sets the tariff to 10 percent, as in an example of +{cite:t}`BassettoHuoRiosRull2026`. + +The Ramsey plan, sustained by the threat of permanent reversion, then switches to $\tau^M$ forever. + +In the organizational equilibrium, the transition map still applies: the next tariff is +$\mathcal{T}(0.1)$, and liberalization resumes from there. + +```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Tariffs after a protectionist shock + name: fig-oe-shock +--- +t_shock, τ_shock, n_plot = 10, 0.1, 30 +τ_O_shock = list(τ_O[:t_shock]) + [τ_shock] +while len(τ_O_shock) < n_plot: + τ_O_shock.append(𝒯(τ_O_shock[-1])) +τ_S_shock = np.r_[τ_R[:t_shock], τ_shock, np.full(n_plot - t_shock - 1, τ_M)] +t_back = np.argmin(np.abs(τ_O - 𝒯(τ_shock))) +print(f"tariff after the shock: {100 * 𝒯(τ_shock):.1f} percent, the undisturbed path's " + f"tariff at period {t_back}") + +fig, ax = plt.subplots() +ax.plot(100 * np.array(τ_O_shock), 'C3-o', ms=4, lw=2, label="organizational equilibrium") +ax.plot(100 * τ_S_shock, 'C0--s', ms=4, lw=2, label="plan sustained by Markov reversion") +ax.plot(np.full(n_plot, 100 * τ_M), 'k:', lw=2, label="Markov equilibrium") +ax.axvline(t_shock, color='gray', ls=':', lw=1) +ax.set_xlabel("period (seven years)") +ax.set_ylabel("tariff (percent)") +ax.legend(loc='upper right', bbox_to_anchor=(1, 0.88)) +plt.show() +``` + +One period after the shock, the organizational tariff is back at the level that the undisturbed path +reached at period 3, so the shock delays liberalization by eight periods, about 56 years, but does not +change where liberalization ends up. + +In the plan sustained by Markov reversion, a one-period disturbance permanently destroys cooperation, a +self-inflicted punishment that later policymakers would want to renegotiate. + +### A quantitative version + +{cite:t}`BassettoHuoRiosRull2026` replace the two-sector static block with a model of the G7 economies +with 30 sectors, input-output linkages, and trade in final and intermediate goods, disciplined by the +World Input-Output Database and KLEMS data. + +Because aggregate capital enters each firm's technology as an externality, the static equilibrium is again +proportional to world capital, and the dynamic policy problem depends on the static block only through +$\chi(\tau)$ and $G(\tau)$, as in {eq}`oe-tariff-payoff`. + +That separability lets them compute $\chi(\tau)$ and $G(\tau)$ by exact-hat algebra {cite}`DekleEatonKortum2008` +from observed trade shares, without separately estimating technologies and tastes. + +They find a Markov tariff of about 12 percent and a best constant tariff of about 1.3 percent, and that +the tariff raises worker welfare mainly by compressing consumption differences across sectors through the +equal rebate of tariff revenue. + +(oe_connections)= +## Connections to other lectures + +Organizational equilibrium adds a fourth equilibrium concept to the three that several lectures in this +section study. + +* {doc}`gradual_trade_agreements` explains gradual liberalization differently, as a feature of efficient + self-enforcing agreements between countries of different size that are sustained by the threat of a + permanent trade war {cite}`BondPark2002`, the kind of cooperation whose fragility to a single + deviation {ref}`oe_tariffs` illustrates. +* {doc}`calvo` computes a Ramsey plan, a Markov perfect equilibrium, and a constrained-to-constant-growth + Ramsey plan in a model of inflation; the last is the analogue of the best constant action, and an + organizational equilibrium converges to it but, because of no delay, does not start there. +* {doc}`calvo_abreu` supports good outcomes in the same model with Abreu's severe stick-and-carrot + punishments {cite}`Abreu`, whereas an organizational equilibrium can use punishments proportional to the + deviation and gives every policymaker the same action payoff. +* {doc}`chang_credible` computes the values of credible plans by iterating on a set-to-set operator in the + manner of {cite:t}`APS1986` and {cite:t}`APS1990`, an iteration that Bassetto, Huo, and Ríos-Rull cannot + use because their symmetry requirement, that every continuation give the same value as the whole + equilibrium, is an equality constraint. +* {doc}`dyn_stack`, {doc}`opt_tax_recur`, and {doc}`chang_ramsey` make Ramsey plans recursive by carrying + promised values or promised private-sector decisions as state variables, which is needed because, as in + the tariff model, later actions are chosen partly to shape earlier private decisions. +* {doc}`dovis_accounting_mf` studies a government that cannot fully commit and whose economy switches + endogenously between monetary-dominant and fiscal-dominant regimes; organizational equilibrium offers + a way to think about how institutions that discipline such a government might be built up over time. +* {doc}`atkeson_1991` and {doc}`tsyrennikov_2013` study lending sustained by the threat of permanent + exclusion into autarky, a punishment of the same permanent kind as reversion to the Markov equilibrium + in {ref}`oe_tariffs`. + +## Exercises + +```{exercise} +:label: oe_ex1 + +In the growth model of {ref}`oe_growth`, the organizational equilibrium converges to the best constant +saving rate $s^B$, which lies below the Ramsey planner's long-run rate $s^R = \alpha\beta$. + +For $\delta \in \{0.5, 0.7, 0.9, 0.95, 0.99\}$, holding $\alpha = 0.4$ and $\beta = 0.96$ fixed, compute +$s^M$, the first organizational saving rate $s_0 = q(s^M; V^\ast)$, $s^B$, and $s^R$, and the numbers of +periods that the organizational path takes to close half and 90 percent of the gap between $s_0$ and +$s^B$. + +How do the long-run gap $s^R - s^B$ and the speed of convergence depend on $\delta$? +``` + +```{solution-start} oe_ex1 +:class: dropdown +``` + +```{code-cell} ipython3 +α_g, β_g = 0.4, 0.96 + +def periods_to_close(fraction, s_start, s_target, V, δ): + "Periods until the organizational path closes the given fraction of the gap." + s, t = s_start, 0 + while s_target - s > (1 - fraction) * (s_target - s_start): + s = q(s, V, α_g, β_g, δ) + t += 1 + return t + +print(" δ s_M s_0 s_B s_R s_R - s_B periods to close 50% and 90%") +for δ_i in [0.5, 0.7, 0.9, 0.95, 0.99]: + s_M_i, s_B_i, s_R_i = benchmarks(α_g, β_g, δ_i) + V_i = action_payoff([s_B_i], s_B_i, α_g, β_g, δ_i) + s_0_i = q(s_M_i, V_i, α_g, β_g, δ_i) + t_50, t_90 = [periods_to_close(f, s_0_i, s_B_i, V_i, δ_i) for f in (0.5, 0.9)] + print(f"{δ_i:5.2f} {s_M_i:7.4f} {s_0_i:7.4f} {s_B_i:7.4f} {s_R_i:7.4f}" + f" {s_R_i - s_B_i:10.5f} {t_50:6d} {t_90:6d}") +``` + +As $\delta$ rises toward one, the present bias weakens, and $s^M$, $s_0$, and $s^B$ all approach the +Ramsey rate $s^R$, but $s^B < s^R$ for every $\delta < 1$. + +Whatever $\delta$, the organizational path closes half of its initial gap in four or five periods but +needs 34 to 46 periods to close 90 percent of it, because $q(\cdot; V^\ast)$ has slope one at $s^B$. + +Convergence is slightly faster when the present bias is weak. + +```{solution-end} +``` + +```{exercise} +:label: oe_ex2 + +The difference equation {eq}`oe-q` with $\bar V = V^\ast$ has a solution for each starting point +$\hat s \in [s^M, s^B]$, and all of them give the first planner the action payoff $V^\ast$. + +1. For a grid of starting points, compute the first planner's gain from sitting out, + $V(s^M, \hat s, s_1, \ldots) - V^\ast$, and confirm that the no-delay condition {eq}`oe-no-delay` holds + exactly when $\hat s \le q(s^M; V^\ast)$. +1. Use {eq}`oe-q` to show that the gain equals $\beta (1 - \delta) \left[ \log\left(1 - q(s^M; V^\ast)\right) - + \log(1 - \hat s) \right]$. + +Hint: compute each path for many periods, and append $s^B$ as the constant tail. +``` + +```{solution-start} oe_ex2 +:class: dropdown +``` + +```{code-cell} ipython3 +def constant_value_path(s_start, n=5000): + s = [s_start] + for t in range(n): + s.append(q(s[-1], V_star)) + return s + +starts = np.linspace(s_M, s_B, 9) +gains = [action_payoff([s_M] + constant_value_path(x), s_B) - V_star for x in starts] +formula = β * (1 - δ) * (np.log(1 - q(s_M, V_star)) - np.log(1 - starts)) +print(f"max |computed gain - formula| = {np.abs(np.array(gains) - formula).max():.1e}") + +fig, ax = plt.subplots() +ax.plot(starts, gains, 'o-', lw=2) +ax.axhline(0, color='k', lw=0.8) +ax.axvline(q(s_M, V_star), color='C3', ls=':', label=r"$q(s^M; V^*)$") +ax.set_xlabel(r"starting saving rate $\hat s$") +ax.set_ylabel("gain from sitting out") +ax.legend() +plt.tight_layout() +plt.show() +``` + +The gain from sitting out is negative for low starting points, zero at $\hat s = q(s^M; V^\ast)$, and +positive above it, where the first planner would rather save at the Markov rate and let its successor start +the sequence. + +For the formula, write $V_t$ for the action payoff of the sequence $s^M, \hat s, s_1, \ldots$ from date $t$ +on, so that $V_1 = V^\ast$. + +Subtracting $\beta V_1$ from $V_0$ as in the derivation of {eq}`oe-q` gives +$V_0 - \beta V^\ast = \log(1 - s^M) + \frac{\delta\alpha\beta}{1 - \alpha\beta} \log s^M - \beta(1 - \delta) +\log(1 - \hat s)$. + +By the definition of $q$, the first two terms on the right side equal +$(1 - \beta) V^\ast + \beta(1 - \delta) \log\left(1 - q(s^M; V^\ast)\right)$. + +Substituting gives $V_0 - V^\ast = \beta (1 - \delta) \left[ \log\left(1 - q(s^M; V^\ast)\right) - +\log(1 - \hat s) \right]$, which is positive exactly when $\hat s > q(s^M; V^\ast)$ and rises with $\hat s$. + +```{solution-end} +``` + +```{exercise} +:label: oe_ex3 + +In the tariff model of {ref}`oe_tariffs`, time inconsistency arises because capitalists' saving responds +to expected returns. + +For $\sigma \in \{0.3, 0.5, 0.7, 1.0\}$, compute the Markov tariff $\tau^M$, the best constant tariff +$\tau^B$, and the first organizational tariff $\mathcal{T}(\tau^M)$. + +Explain why the three coincide when $\sigma = 1$. +``` + +```{solution-start} oe_ex3 +:class: dropdown +``` + +```{code-cell} ipython3 +print(" σ τ_M (%) τ_B (%) T(τ_M) (%)") +for σ_i in [0.3, 0.5, 0.7, 1.0]: + model_i = create_tariff_model(σ=σ_i) + τ_M_i, τ_B_i = markov_tariff(model_i), best_constant_tariff(model_i) + H_B_i = H(τ_B_i, s_const(τ_B_i, model_i), model_i) + τ_0_i = transition(τ_M_i, τ_B_i, H_B_i, model_i) + print(f"{σ_i:5.1f} {100 * τ_M_i:8.2f} {100 * τ_B_i:8.2f} {100 * τ_0_i:10.2f}") +``` + +The Markov tariff does not depend on $\sigma$, because a Markov policymaker takes the saving rate as given. + +The best constant tariff rises with $\sigma$, because the saving rate responds less to the return to +capital, and so does the first organizational tariff. + +With $\sigma = 0.3$, the best constant tariff is at the corner $\tau^B = 0$. + +With $\sigma = 1$, capitalists save the constant fraction $\beta$ of their wealth whatever the return, so +expected tariffs do not affect saving, there is no time inconsistency, and the Markov, best constant, and +organizational tariffs coincide. + +```{solution-end} +```