RBC Models of Small Open Economies

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1 RBC Models of Small Open Economies Enrique G. Mendoza University of Pennsylvania, NBER & PIER Based on: The Terms of Trade, the Real Exchange Rate, and Economic Fluctuations, International Economic Review, 36(1), Feb. 1995,

2 General motivation Use this paper as an example of a canonical RBC-SOE model with incomplete markets Discuss issues specific to the topic of the paper (the role of TOT in driving business cycles) but also focus on methodological issues: Second approach to study determination of wealth dynamics (Uzawa-Epstein preferences) Solution & calibration of multisector SOE model Role of wealth effects introduced by incomplete markets

3 Layout of the lecture 1. Review goals and 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

4 1. GOALS AND LITERATURE PERSPECTIVE

5 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 < s papers by OSR Deterministic intertemporal models. Positive relationship if TOT shock is transitory and lower 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. (2016) Role of TOT in business cycles: news v. unexpected

6 Complete Markets-BKK models Country-specific TOT shocks are fully insurable with state-contingent assets, and thus persistence of TOT shocks is irrelevant Wealth and its cross-country distribution are timeand state-invariant Differences in preferences and underlying shocks determine eq. dynamics of both 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

7 Incomplete Markets Approach Wealth effects because of 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.

8 What is the paper about? 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

9 2. CROSS COUNTRY STYLIZED FACTS OF BUSINESS CYCLES AND TERMS-OF-TRADE FLUCTUATIONS

10 The HLM Effect in Data

11 The HLM Effect in Data (Caputo, Irarrazabal & Romero, 2013)

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24 3. MULTISECTOR RBC-SOE MODEL WITH INCOMPLETE MARKETS

25 Preferences Model-based approach to obtain well-defined stationary NFA distribution by adopting Uzawa- Epstein Stationary Cardinal Utility Infinitely-lived representative household consumes fours goods: Nontradables, n Exportables, x Importables, f Leisure, l

26 Stationary Cardinal Utility Endogenous rate of time preference introduces impatience effect, i.e. (rate of time preference is increasing in past consumption) Epstein JET 1983 provides axiomatic theory behind SCU and existence theorem for SCU as only expected utility function consistent with axioms. Also proves conditions for existence of unique, invariant distribution of wealth for linear and neoclassical storage technologies

27 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. [.]

28 Technology and Asset Markets

29 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

30 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

31 Methodology 1. Characterize competitive eq. as planner s problem 2. Represent equilibrium in recursive form as solution to 3-dimesional stochastic DP problem 3. Simplify uncertainty with first-order Markov chains, assuming also symmetry and Simple Persistence 4. Design solution algorithm based on VFI 5. Calibrate model to SS representative of industrialized (developing) countries 6. Solve giving as input observed TOT processes and TFP shocks that match basic business cycle facts 7. Examine equilibrium co-movements (benchmark simulations, sensitivity analysis, impulse responses).

32 4. SOCIAL PLANNER S PROBLEM

33 Planner s problem in standard form is the multiplier on ( 4 ) is the multiplier on ( 5 ) is the multiplier on ( 6a ) is the multiplier on ( 6b )

34 First-order condition for exportables consumption X

35 First-order conditions for importables, nontradables and leisure

36 First-order conditions for factor allocations and Euler equations

37 Simplifying First-Order Conditions Define Lifetime marginal utilities as of date t Impatience effect Note that because of constant-elasticity functional forms, impatience effects cancel from marginal rates of substitution across x, f, n, l

38 Simplified optimality conditions

39 Simplified optimality conditions

40 Recursive Representation Express endogenous variables as recursive functions of states, where, and f as control variable 1. Use period utility (2), time preference function (3) and MRS between x and f (8) to obtain 2. Use nontradables market clearing (5), MRS between l and n (10) and (13) to get as solution to:

41 Recursive Representation 3. 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

42 Recursive Planner s Problem Using (8)-(11) we obtain:

43 5. NUMERICAL SOLUTION METHOD

44 Simplifying Uncertainty Two-point, first-order, symmetric Markov processes and perfectly correlated TFP shocks Markov processes under Simple Persistence: is long run probability of ( )

45 Simple Persistence Rule Long run probability of state j Simple persistence parameter Transition prob between states i and j

46 Simple Persistence Rule Vector of realizations: Transition probability matrix

47 Unconditional Moments of Shocks Symmetry conditions: Expected values are zero:

48 Unconditional Moments of Shocks Variances and std. devs.: AR(1) s are common and equal to :

49 Unconditional moments of shocks Contemporaneous correlation: Given from the data we can check if SP is reasonable and we can calibrate the Markov chain

50 Discrete Recursive Planner s Problem Recover recursive planner s problem: Use discrete grids for to define the state space of dimensions JxHX4 Capital, NFA grids can be centered at det. steady state

51 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

52 Standard VFI Solution Method

53 Computing stochastic steady state The stochastic steady state is the unique, invariant stationary distribution of capital, NFA and shocks Solved by converging on the law of motion of conditional probabilities

54 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:

55 6. CALIBRATION

56 Calibration for Industrial (Developing) Country Baselines TFP and TOT shocks: Preferences: Technology

57 Using Deterministic Steady State for Calibration Parameters: these G7 averages: plus 1. from KHS, 1975 point estimate 2. IMF data, UNCTAD Endogenous: Solve for the endogenous variables using det. SS simultaneous equation system

58 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

59 Steady State Equation System

60 Steady State Equation System

61 7. RESULTS OF THE QUANTITATIVE ANALYSIS

62 Can the model explain the HLM Effect? is low and positive in the model, although a bit lower than in the data v in G7 data v in DC 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

63 Matching the cross section of the HLM effect Take IC and DC benchmarks Solve the model 30 times using actual AR(1) s of TOT for each of 30 countries in Table 1 Regress simulated on AR(1) coefficients and plot simulated line in Figure 2 Low and positive independent of TOT autocorrelations are consistent with DSGE model of SOE with incomplete markets

64 HLM correlations in data and model

65 What determines the TB-TOT correlation Sensitivity analysis varying IC benchmark parameters to DC benchmark one at a time Higher AR(1) of TOT, caeteris paribus, implies lower, as in OSR models but also changes with the other parameters, and it is particularly sensitive to changes in variance-covariance structure of the shocks.

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67 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:

68 Lessons from Business Cycle Moments TOT shocks explain 49% of G7 and 56% of DC output fluctuations Model explains 40-50% of the variability of RER and is consistent with procyclical RER Model produces large interest rate differentials (3.6% for ICs, 4.6% for DCs), with low and positive correlations with RER Uniformity of business cycles across DCs and ICs is replicated for the most part (except TB/Y) Model matches several qualitative features, but quantitatively it is not a perfect match

69 Impulse response functions: Aggregates (measured at import prices) Terms of trade Productivity

70 Impulse response functions: Sectoral Terms of trade Productivity

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