RBC Models of Small Open Economies

Enrique G. MendozaUniversity of Pennsylvania, NBER & PIER

Based on: “The Terms of Trade, the Real Exchange Rate, and Economic Fluctuations,” International Economic Review, 36(1), Feb. 1995, 101-137.

General motivation• An example of canonical RBC-SOE model with

incomplete markets

• Discuss issues specific to the topic of the role of TOT in driving business cycles

• …but also focus on methodological issues:– Determination of wealth dynamics using Uzawa-

Epstein preferences– Solution & calibration of multisector SOE model– Role of wealth effects under incomplete markets

Layout of the lecture

1. Review of related literature

2. Document cross country business cycle facts

3. Propose multi-sector RBC-SOE model

4. Characterize recursive social planner’s problem

5. Describe solution method

6. Calibrate the model to IC and DC data

7. Examine quantitative implications

1. LITERATURE REVIEW

Literature on the relationship between the Terms of Trade and Trade balance

• 1950s papers by HLM– Corr(TB,TOT)>0 because of propensity to import < 1

• 1980s papers by OSR– Deterministic intertemporal models.– Positive co-movement weakens as persistence rises

• 1990s Backus JIE, BKK IRBC papers– Complete markets, DSGE Modeling – Persistence of TOT does not matter

• 2010s SGU (2016), Zeev et al. (2017)– Role of TOT in business cycles: news v. unexpected

using DEIR function

Complete Markets-BKK models• Country-specific TOT shocks are fully insurable

with state-contingent assets, hence persistence of TOT shocks is irrelevant

• Wealth and its cross-country distribution are time-and state-invariant (history independence)

• Differences in preferences and shocks determine eq. dynamics of TOT and TB.1. TOT does not fluctuate nearly enough in the models.2. ρ(c,c*) << ρ(y,y*) in the data!

• Several stylized facts at odds w. these models– Excessive consumption corrs., TOT, RER variability

Incomplete Markets Approach

• Wealth effects due to state-contingent wealth– Countries with longer spells of high TOT are wealthier– Self insurance against TOT shocks

• Persistence of TOT shocks matters, but at equilibrium it can be offset by differences in other properties of shocks or the structure of preferences and technology.

What is done in Mendoza IER?• Analysis of cross-country business cycle facts,

particularly the relationship between TB & TOT– Corr(TOT,TB) favors Harberger-Laursen-Metzler

(correlations are generally positive, but low).– Does not seem to favor Obstfeld-Svensson-Razin:

corr(TB,TOT) unrelated to ρ(TOT)– DCs are more volatile than ICs, but high degree of

uniformity in persistence and co-movement

• Analysis of cyclical implications of TOT & TFP shocks in an RBC-SOE model– Model can explain cyclical dynamics of TB & TOT– TOT shocks are important drivers of business cycles

2. CROSS COUNTRY STYLIZED FACTS OF BUSINESS CYCLES

AND TERMS-OF-TRADE FLUCTUATIONS

The HLM Effect in 1955-1990 Data

The HLM Effect in 1955-2010 Data(Caputo, Irarrazabal & Romero, 2013)

3. MULTISECTOR RBC-SOE MODEL WITH INCOMPLETE

MARKETS

Preferences: Stationary Cardinal Utility• Theory-based approach to obtain well-defined stationary

NFA distribution using Epstein’s Stationary Cardinal Utility

– Infinitely-lived representative agent consumes four goods: Exportables, x, importables, f, nontradables, n, & leisure, l

– Endogenous rate of time preference that is increasing in past consumption introduces “impatience effect”

– Epstein JET 1983: axiomatic theory behind SCU proving that SCU is the only expected utility function consistent with axioms. Also proves conditions for existence of unique, invariant distribution of wealth for linear and neoclassical technologies

Functional Forms for Preferences

• is implied by Epstein’s SCU analysis.• elasticity of substitution between &

• share of consumption of in (unitary elasticity)• Frisch elasticity of labor supply• semi-elasticity of rate of time preference w.r.t. [.]

Technology and Asset Markets

– mean terms of trade– stationary terms of trade shock– stationary productivity shock to exportables– capital stock allocated to exportables– inelastic labor allocated to exportables– share of labor income in exportables– stationary productivity shock to importables– capital stock allocated to importables– inelastic labor allocated to importables– share of labor income in importables– aggregate, homogeneous capital in tradables– depreciation rate

– adjustment cost curvature– NFA in non-contingent, one period debt– world interest rate– stationary TFP shock to nontradables industry– inelastic capital stock in nontradables industry– labor supply allocated to nontradables industry– share of labor income in nontradables industry

• Model allows consumption & production of all goods• Whether x is exported and f imported is an

equilibrium outcome• Note assumptions about factor mobility

Methodology

1. Characterize competitive eq. as planner’s problem2. Represent equilibrium in recursive form as solution to

3-dimesional stochastic DP problem3. Simplify uncertainty with first-order Markov chains,

assuming symmetry and “Simple Persistence”4. Design solution algorithm5. Calibrate model to SS representative of industrialized

(developing) countries6. Solve giving as input observed TOT processes and

TFP shocks that match data7. Examine equilibrium co-movements (benchmark

simulations, sensitivity analysis, impulse responses).

4. SOCIAL PLANNER’S PROBLEM

Sequential planner’s problem

• is the multiplier on ( 4 )

• is the multiplier on ( 5 )

• is the multiplier on ( 6a )

• is the multiplier on ( 6b )

First-order condition for exportables consumption

X

First-order conditions for importables, nontradables and leisure

First-order conditions for factor allocations and Euler equations

Simplifying First-Order Conditions

• Define

• Because of constant-elasticity functional forms, impatience effects cancel from marginal rates of substitution across x, f, n, l !

Lifetime marginal utilities as of date t

“Impatience effect”

Simplified optimality conditions

Simplified optimality conditions

Effects of an increase in TOT1. Consumption of x becomes relatively more

expensive (unitary elasticity x v. f, CES T v. n)

2. Income/wealth effects because of higher value of exports in terms of imports

3. Immediate reallocation of capital from production of f to production of x

4. Increase in investment because of persistence of TOT improvement

5. Effect on TB is the net result of above effects plus self-insurance incentives on NFA

Recursive Representation• Express endogenous variables as recursive functions of

states , where , and of f as a control variable

1. Use MRS between x and f (eq. (8)) to obtain

2. Use nontradables market clearing (5), MRS between land n (10) and (13) to get as solution to:

Recursive Representation3. Use nontradables market clearing (5), MRS between n

and f (9), and (13) and (14) to obtain and

4. Use equalization of marginal products of sectoral capital (11) and to get capital allocations

Recursive Planner’s Problem• Given , , ,

, , the recursive planner’s problem is:

5. NUMERICAL SOLUTION METHOD

Simplifying Uncertainty

• Two-point, first-order, symmetric Markov processes with perfectly correlated TFP shocks

• Implications of Simple Persistence rule:

is long run probability of ( )

Simple Persistence Rule

Long run probability of state j

Simple persistence parameter

Transition prob between states i and j

Simple Persistence Rule• Vector of realizations:

• Transition probability matrix

Unconditional Moments of Shocks• Symmetry conditions:

• Expected values are zero:

• Homework: show that these are the other moments

• Given from data we can check if SP is reasonable and calibrate the Markov chain

Discrete Recursive Planner’s Problem• Rewrite recursive planner’s problem:

• Use discrete grids for to define the state space of dimensions JxHX4– has 4 nodes of shock

quadruples – Capital, NFA grids can be centered at det. steady state

Value Function Iteration Algorithm• Standard VFI: Make initial guess of the value

function and iterate on the Bellman equation:

s.t.

within the set is o.k.)

• Tips:– Decision rules converge faster– Faster if combined with policy function iteration

But why bother, use FiPIt algorithm!• Replace endogenous discounting w. standard preferences

1. Define a quasi pricing function: 𝑞𝑞 𝐾𝐾,𝐴𝐴, 𝜆𝜆 = 1 + 𝜙𝜙 𝐾𝐾′ 𝐾𝐾,𝐴𝐴, 𝜆𝜆 − 𝐾𝐾

2. Start with conjectures �𝑞𝑞 𝐾𝐾,𝐴𝐴, 𝜆𝜆 , �̂�𝐴′ 𝐾𝐾,𝐴𝐴, 𝜆𝜆

3. Quasi pricing function yields implied �𝐾𝐾′ 𝐾𝐾,𝐴𝐴, 𝜆𝜆

4. Equilibrium conditions then yield 𝑓𝑓 𝐾𝐾,𝐴𝐴, 𝜆𝜆

5. Use FiPIt in bonds Euler eq. to solve for new 𝑓𝑓 𝐾𝐾,𝐴𝐴, 𝜆𝜆 , and then resource constraint to get new 𝐴𝐴′(𝐾𝐾,𝐴𝐴, 𝜆𝜆).

6. Use FiPIt in capital Euler eq. to solve for new 𝑞𝑞 𝐾𝐾,𝐴𝐴, 𝜆𝜆

7. Iterate to convergence on |𝑞𝑞 𝐾𝐾,𝐴𝐴, 𝜆𝜆 − �𝑞𝑞 𝐾𝐾,𝐴𝐴, 𝜆𝜆 | and |𝐴𝐴′ 𝐾𝐾,𝐴𝐴, 𝜆𝜆 − �̂�𝐴′ 𝐾𝐾,𝐴𝐴, 𝜆𝜆 |

Computing stochastic steady state

• Stochastic steady state: unique, invariant stationary distribution of capital, NFA and shocks

• Solved by converging on the law of motion of conditional probabilities

Computing stochastic steady state

• Moments of all of the model’s variables can be computed using the limiting distribution of states

• Impulse response functions can be computed by calculating the OLS coefficients of the system:

6. CALIBRATION

Calibration of TOT and TFP ShocksIndustrial (Developing) Country

• TFP: from Stockman & Tesar (AER, 1994)– AR(1) of and at 0.632 and 0.154 with 0 off-

diagonal elements (supports simple persistence)

• TOT: G7 means in Table 1

• Using ST & Table 1: , , and chose mid-point

Calibration of Preferences

• Using T/NT data from Kravis, Heston and Summers (1982, 1983) “Income Comparison Project,” run this regression:

• Assumed labor/leisure allocations:

• CRRA parameter in standard range

• set so that deterministic steady state matches average and from G7 data

Calibration of Production

• Combine data from KHS, ST, World Bank &OECD– nontradables labor share in G7 (ST)– G7 mean labor share in manufacturing (WB)

(net exporters of manufactures=> f=manufactures)– follows from , (ST),

(manufacturing in GDP, WB), and:

• normalization, from Mendoza (AER 1991)

Using Deterministic Steady State for Calibration

• Parameters: plus these G7 averages:1. from KHS, 1975 point estimate

2. IMF data, 1965-91

3. UNCTAD

• Endogenous:

• Solve for the endogenous variables using det. SS simultaneous equation system

Steady State Equation System

• Capital allocations solve a three-equation block recursive system:

• The rest of the variables are obtained from a 9-equation nonlinear simultaneous eq. system

Steady State Equation System

Steady State Equation System

7. RESULTS OF THE QUANTITATIVE ANALYSIS

Can the model explain the HLM Effect?

• is low and positive in the model, although a bit lower than in the data

– G7: 0.185 (model) v. 0.241 (data)– DC: 0.076 (model) v. 0.317 (data)

• Structural parameter differences explain lack of co-movement between and AR(1) of TOT.

– Even though AR(1) of > AR(1) of ,is slightly higher

Matching the cross section of the HLM effect

1. Take IC and DC benchmarks

2. Solve the model 30 times using actual AR(1)’s of TOT for each of 30 countries in Table 1

3. Regress simulated on AR(1) coefficients and plot simulated line in Figure 2

• Low and positive independent of TOT autocorrelations are consistent with RBC model of SOE with incomplete markets!

HLM correlations in data and model

What determines the TB-TOT correlation

• Sensitivity analysis varying IC benchmark parameters to DC benchmark one at a time

• Higher AR(1) of TOT, everything else constant, implies lower , as in OSR models!

• But also changes with other parameters, and is particularly sensitive to changes in variance-covariance structure of the shocks (which alter wealth effects)

• Preference parameters alone matter much less!

Business Cycle Moments

• Moments for IC/DC benchmarks in Tables 7, 8– Valued at import prices and at aggregate prices

• CPI: Given CES, there is an expenditure function PC = E where

– Used to compute RER and domestic interest rate, which is risk-free return in units of C:

– Explains main finding of Ayres, Hevia & Nicolini(2017): TOT contribute to explain RER variability, even in advanced economies

Lessons from Business Cycle Moments 1. TOT shocks explain 49% of G7 and 56% of DC

output fluctuations

2. Model explains 40-50% of the variability of RER and is consistent with procyclical RER

3. Model produces large interest rate differentials (3.6% for ICs, 4.6% for DCs), with low and positive correlations with RER

4. Uniformity of business cycles across DCs and ICs is replicated for the most part (except TB/Y)

5. Model matches several qualitative features, but quantitatively it is not a perfect match

Impulse response functions: Aggregates(measured at import prices)

Terms of trade shock Productivity shock

Impulse response functions: Sectoral

Terms of trade shock Productivity shock