Graduate Macroeconomics - Econ 551

Transcription

1 Graduate Macroeconomics - Econ 551 Tack Yun Indiana University Seoul National University Spring Semester January 2013 T. Yun (SNU) Macroeconomics 1/07/ / 32

2 Business Cycle Models for Emerging-Market Economies 1 Canonical real business cycle models Log-linear approximation to equilibrium conditions Solution to dynamic linear models 2 Real business cycle models for emerging-market economies Two real business cycle models for emerging-market economies Sources of business cycles in EME RBC models Permanent and temporary shocks to productivity country-premium shocks Preference shocks Spending shocks 3 Extensions of real business cycle models for emerging-market economies Imperfect information regarding technology shocks Labor-market search Introduction of working-capital loans T. Yun (SNU) Macroeconomics 1/07/ / 32

3 Real Business Cycle Model 1 Models of Business Cycles (1988) by Robert E. Lucas, Jr. pp The Kydland and Prescott model is a highly simplified, competitive system, in which a single good is produced by labor and capital with a constant-returns-to-scale technology. 3 All consumers are assumed to be infinitely-lived and identical. 4 The only shocks to the system are exogenous, stochastic shifts in the production technology. 5 Can specific parametric descriptions of technology and preferences be found such that the movements induced in output, consumption, employment and other series in such a model by these exogenous shocks resemble the time series behavior of the observed counterparts to these series in the postwar, US economy? T. Yun (SNU) Macroeconomics 1/07/ / 32

4 Technology and Production Factors Markets 1 The production technology is exp(a t )F (K t, N t ), where F is homogenous of degree one in (K, N). 2 Output is divided into consumption C t and gross investment I t. Y t = C t + I t 3 Capital evolves according to a linear accumulation technology: K t+1 = I t + (1 δ)k t 4 Households own all factors of production, renting them to profit-maximizing firms each period at wages and capital rentals W t and R t. Markets for production factors are perfectly competitive. T. Yun (SNU) Macroeconomics 1/07/ / 32

5 Household 1 We begin with the optimization problem of households. 2 Preferences β t E 0 [u(c t, 1 n t )] t=0 where E 0 denotes the expectation conditional on the information set at period 0. 3 Budget constraint at period t c t + k t+1 W t n t + (R t + 1 δ)k t 4 At period 0, each household chooses a plan on current and future levels of consumption, hours worked, and stocks of capital by maximizing its welfare subject to a sequence of budget constraints, taking as given {W t, R t } t=0 and k 0. T. Yun (SNU) Macroeconomics 1/07/ / 32

6 Deterministic Plan versus Stochastic Plan 1 A deterministic plan permits only one sequence of numbers for each choice variable such as consumption, hours worked, and capital stock. 2 There is only one path for current and future levels of total factor productivity. 3 Rational expectation requires perfect foresight for the time path of the total factor productivity. 4 A stochastic plan allows for the possibility that the future value of the total factor productivity can be realized randomly within a set of distinctly different values. 5 We assume that all households and firms know the true stochastic law of motion for the total factor productivity. T. Yun (SNU) Macroeconomics 1/07/ / 32

7 Formulation of a Typical Household s Decision Problem 1 The state facing each household at each date is described by its own holdings of capital (= k), the capital stock in the economy as a whole (= K), and the current technology shock (= a). 2 The definition of state vector for each household: s t = {k t, K t, a t }. 3 The definition of the aggregate state vector: S t = {K t, a t }. 4 The history of states: s t = (s 0, s 1,, s t ). 5 The representative household at period 0 chooses its stochastic plan by solving the following problem: max {{c(st),n(st),k(s t)}} t=0 t=0 s t βt prob(s t s 0 ){u(c(s t ), 1 n(s t ))+ λ(s t )(W (S t )n(s t ) + (R(S t ) + 1 δ)k(s t 1 ) c(s t ) k(s t ))} T. Yun (SNU) Macroeconomics 1/07/ / 32

8 Optimization Conditions of Household s Decision Problem 1 Consumption u c (c t, 1 n t ) = λ t 2 Labor u l (c t, 1 n t ) = λ t W t 3 Capital Stock λ t = βe t [λ t+1 (R t δ)] T. Yun (SNU) Macroeconomics 1/07/ / 32

9 Equilibrium Conditions 1 Symmetry of Individual Households: K t = k t ; N t = n t ; C t = c t. 2 Labor Market Equilibrium u l (C t, 1 N t ) = u c (C t, 1 N t ) exp(a t )F n (K t, N t ) 3 Capital Stock Evolution 1 = βe t [ u c(c t+1, 1 N t+1 ) (exp(a t+1 )F k (K t+1, N t+1 ) + 1 δ)] u c (C t, 1 N t ) 4 Goods Market Equilibrium 5 Solutions C t + K t+1 (1 δ)k t = exp(a t )F (K t, N t ) C t = C(K t, a t ); N t = N(K t, a t ); K t+1 = K(K t, a t ) T. Yun (SNU) Macroeconomics 1/07/ / 32

10 Log-Linear Approximation 1 Original (nonlinear) equilibrium conditions are replaced by a system of linear equations, which are rewritten in terms of each variable s logarithmic deviation from its (deterministic) steady-state value. 2 Definition of logarithmic deviation: Ĉ t = log C t log C where C is the value of C t at the deterministic steady state. 3 Relation between C t and Ĉ t : C t = exp(ln C t ) = C exp(ln C t ln C) = C exp(ĉ t ) 4 First-order Taylor expansion of exp(ĉ t ): exp(ĉ t ) 1 + exp(0)ĉ t T. Yun (SNU) Macroeconomics 1/07/ / 32

11 Equilibrium Conditions under Logarithmic Utility and Cobb-Douglas Production Functions 1 Utility Function and Production Function: u(c t, 1 N t ) = ln C t + b n (1 N t ); Y t = exp(a t )K α t N 1 α t 2 Labor Market Equilibrium b n 3 Capital Stock Evolution C t = (1 α) exp(a t )( K t ) α 1 N t N t C t 1 = βe t [ (α exp(a t+1 )( N t+1 ) 1 α + 1 δ)] C t+1 K t+1 4 Goods Market Equilibrium C t + K t+1 (1 δ)k t = exp(a t )K α t N 1 α t T. Yun (SNU) Macroeconomics 1/07/ / 32

12 Example of Log-Linear Approximation to the Holdings of Capital Stock 1 Express the original equilibrium condition in terms of logarithmic deviations exp( Ĉt) = βe t [exp( Ĉt+1)(R exp(ˆr t+1 ) + 1 δ)] 2 Find a term-by-term expression of the resulting equation exp( Ĉ t ) = E t [βr exp( Ĉ t+1 ) exp(ˆr t+1 ) + β(1 δ) exp( Ĉ t+1 )] 3 Apply the first-order approximation to each term Ĉ t = E t [βr( Ĉ t+1 + ˆR t+1 ) β(1 δ)ĉ t+1 ] 4 Simplify the approximated equation by using the corresponding steady-state equilibrium condition Ĉ t = E t [ Ĉ t+1 + βr ˆR t+1 ] T. Yun (SNU) Macroeconomics 1/07/ / 32

13 Example of Log-Linear Approximation to the Holdings of Capital Stock 1 Express the original equilibrium condition in terms of logarithmic deviations exp( Ĉt) = βe t [exp( Ĉt+1)(R exp(ˆr t+1 ) + 1 δ)] 2 Find a term-by-term expression of the resulting equation exp( Ĉ t ) = E t [βr exp( Ĉ t+1 ) exp(ˆr t+1 ) + β(1 δ) exp( Ĉ t+1 )] 3 Apply the first-order approximation to each term Ĉ t = E t [βr( Ĉ t+1 + ˆR t+1 ) β(1 δ)ĉ t+1 ] 4 Simplify the approximated equation by using the corresponding steady-state equilibrium condition Ĉ t = E t [ Ĉ t+1 + βr ˆR t+1 ] T. Yun (SNU) Macroeconomics 1/07/ / 32

14 Example of Log-Linear Approximation to Labor-Market Equilibrium Conditions 1 Express the original equilibrium condition in terms of logarithmic deviations exp(ĉ t ) = (1 N) 1 exp(ŵ t )(1 N exp( ˆN t )) 2 Find a term-by-term expression of the resulting equation exp(ĉt) = exp(ŵ t ) 1 N N exp(ŵ t + ˆN t ) 1 N 3 Apply the first-order approximation to each term Ĉ t = 1 1 N Ŵt N 1 N (Ŵ t + ˆN t ) 4 Simplify the approximated equation by using the corresponding steady-state equilibrium condition N 1 N ˆN t = Ŵt Ĉt; Ŵ t = a t + α( ˆK t ˆN t ) T. Yun (SNU) Macroeconomics 1/07/ / 32

15 Log-Linear Approximation to the Equilibrium Conditions 1 Labor Market Equilibrium 2 Capital Stock (α + N 1 N ) ˆN t = Ĉ t + α ˆK t + a t Ĉ t = E t [ Ĉ t+1 + βr(a t+1 + (1 α)( ˆN t+1 ˆK t+1 ))] 3 Goods Market Equilibrium C Y Ĉt + K Y ( ˆK t+1 (1 δ) ˆK t ) = a t + α ˆK t + (1 α) ˆN t T. Yun (SNU) Macroeconomics 1/07/ / 32

16 Reduced Form 1 Labor Market Equilibrium + Goods Market Equilibrium ˆK t+1 = a kk ˆK t a kc Ĉ t + a ka a t 2 Capital Stock + Goods Market Equilibrium E t [Ĉ t+1 ] = a ck Ĉ t + a cc Ĉ t + a ca a t 3 Logarithmic level of the aggregate productivity a t = ρ a a t 1 + ɛ a,t T. Yun (SNU) Macroeconomics 1/07/ / 32

17 Method of Undetermined Coefficients 1 Guess linear decision rules for consumption and capital stock. Ĉ t = φ k ˆK t + φ k a t ; ˆK t+1 = ϕ k ˆK t + ϕ k a t 2 Verify that your guess is correct. We do this by substituting these two rules into equilibrium conditions and obtaining equations for coefficients of decision rules. We then solve the resulting equations for coefficients. 3 We have a quadratic equation for φ k and a linear equation for φ a given a value of φ k : a kc φ 2 k (a kk a cc )φ k + a ck = 0; φ a = a ca + φ k (a kc a ka ) ρ a a cc φ k a ka T. Yun (SNU) Macroeconomics 1/07/ / 32

18 Decision Rules for Capital Stock, Employment, and Output 1 Capital Stock ϕ k = a kk a kc φ k ; ϕ a = a ka a kc φ a 2 Employment ˆN t = α φ k α + N/(1 N) ˆK α φ a t + α + N/(1 N) a t 3 Output Ŷ t = (α + (1 α)(α φ k) α + N/(1 N) ) ˆK t + (1 + (1 α)(α φ a) α + N/(1 N) )a t T. Yun (SNU) Macroeconomics 1/07/ / 32

19 Construction of State-Space Representation 1 Write the law of motion for the state vector ( ) ( ) ( ) ( ˆKt ϕk ϕ = a ˆKt ρ a 1 a t where ϕ k and ϕ a are defined as a t 1 ) ɛ a,t ϕ k = a kk a kc φ k ; ϕ a = a ka a kc φ a 2 Express endogenous variables in terms of state variables Ŷ t α + (1 α)(α φ k) α+n/(1 N) 1 + (1 α)(α φa) ( α+n/(1 N) Ĉ t ˆKt = φ k φ a a t ˆN t α φ k α+n/(1 N) α φ a α+n/(1 N) ) T. Yun (SNU) Macroeconomics 1/07/ / 32

20 Canonical Real Business Cycle Model of Small Open Economies 1 The Mendoza model (1991) is a highly simplified, competitive system, in which a single good is produced by labor and capital with a constant-returns-to-scale technology. 2 All risk-averse consumers are assumed to be infinitely-lived and identical. 3 The only shocks to the system are exogenous, stochastic shifts in the production technology. 4 There is non-contingent international one-period bond and the asset market is incomplete. T. Yun (SNU) Macroeconomics 1/07/ / 32

21 Key Features of the Canonical Small Open RBC Model 1 The production technology is exp(a t )Kt 1 α Nt α, where a t is given by 2 The social resource constraint is 3 Accumulation of international debt a t = ρ a a t 1 + ɛ a,t. Y t = C t + I t + NX t. q t B t+1 = B t NX t 4 Capital evolves according to a linear accumulation technology with a quadratic adjustment costs: K t+1 = I t + (1 δ)k t + φ 2 (K t+1/k t exp(µ g )) 5 There is no role of permanent shocks to productivity. T. Yun (SNU) Macroeconomics 1/07/ / 32

22 Empirical Regularities of Emerging-Market Business Cycles Emerging-Market Economies Advanced Economies σ(y ) σ( Y ) ρ(y ) ρ( Y ) σ(c)/σ(y ) σ(tb/y ) ρ(tb/y, Y ) T. Yun (SNU) Macroeconomics 1/07/ / 32

23 Motivation for RBC models for Emerging-Market Economies 1 Canonical RBC models of small open economies are not consistent with empirical stylized facts for emerging-market economies. 2 For example, canonical RBC models of small open economies generate pro-cyclical movements of the trade balance and smooth fluctuations of consumption. A positive temporary shock to TFP increases consumption but less than output, thus leading to a surplus in the trade balance. The standard deviation of consumption is less than that of output. The trade balance is procyclical (or acyclical) in models with Cobb-Douglas preferences or weakly counter-cyclical in models with GHH preferences. T. Yun (SNU) Macroeconomics 1/07/ / 32

24 Key Features of Aguiar and Gopinath Model 1 The production technology is exp(a t )F (K t, Γ t N t ), where F is homogenous of degree one in (K, N). 2 Output is divided into consumption C t and gross investment I t. Y t = C t + I t + NX t 3 Accumulation of international debt q t B t+1 = B t NX t 4 Capital evolves according to a linear accumulation technology with a quadratic adjustment costs: K t+1 = I t + (1 δ)k t + φ 2 (K t+1/k t exp(µ g )) 5 Elasticity of the interest rate to changes in indebtedness r t = r + ψ(exp(b t+1 /Γ t b) 1) T. Yun (SNU) Macroeconomics 1/07/ / 32

25 Solow Residuals 1 Specification of production function Cobb-Douglas production function Permanent technology shocks Y t = exp(a t )K 1 α t (Γ t N t ) α. Γ t = exp(g t )Γ t 1, g t = (1 ρ g )µ g + ρ g g t 1 + ɛ g,t Temporary technology shocks 2 Solow residuals a t = ρ a a t 1 + ɛ a,t sr t = a t + α log Γ t 3 Random-walk component and transitory component of Solow residuals sr t = τ t + s t τ t = τ t 1 + αµ g + αρ g ɛ g,t 1 ρ g T. Yun (SNU) Macroeconomics 1/07/ / 32

26 Wealth Effect of Labor Supply 1 Cobb-Douglas Utility Function u t = (C 1 γ t (1 L t ) 1 γ ) 1 σ /(1 σ) 0 < γ < 1 Labor Supply Curve 1 γ γ C t 1 N t = w t 2 Greenwood, Hercowitz and Huffman Preferences u t = (C t τγ t 1 L ν t ) 1 σ /(1 σ) Labor Supply Curve τνγ t 1 N ν 1 t = w t T. Yun (SNU) Macroeconomics 1/07/ / 32

27 Permanent Shocks to Productivity and Relative Consumption Volatility 1 Euler Equation MU c,t = β(1 + r t )E t [MU c,t+1 ] 2 The log-linearization of the Euler Equation around the deterministic steady state leads to the following equation for the ratio of the aggregate consumption to output Ĉ t Ŷ t = 1 E t [ˆR t+k ] + E t [ ˆQ t+k ] γ where ˆQ t is defined as k=0 k=0 ˆQ t = 1 γ {(σ 1)(1 γ)ˆl t (σ(1 γ) 1)Ŷ t } 3 ˆQt depends on the calibration of parameters: ˆQ t = Ŷt when γ = 1 ˆQ t = ˆL t when γ = 1/2 and σ = 2. T. Yun (SNU) Macroeconomics 1/07/ / 32

28 Long-Sample versus Short-Sample for Argentina and Mexico Argentian Mexico σ( Y ) L S ρ( Y ) L S σ(tb/y ) L S ρ(tb/y ) L S T. Yun (SNU) Macroeconomics 1/07/ / 32

29 Autocorrelations of Trade Balance in RBC models of Small Open Economies 1 The autocorrelations of the trade balance predicted by the AG-type RBC model are close to unity, indicating that the model s trade balance-to-output ratio behaves as a near random walk. 2 By contrast, the empirical autocorrelation function takes a value slightly below 0.6 at order one and then declines quickly toward zero. 3 The near random-walk behavior of the trade balance-to-output ratio in the AG-type RBC model is not a consequence of the presence of permanent productivity shocks. 4 Garcia-Cicco, Pancrazi, and Uribe (2010) argue that one way to eliminate the near random-walk behavior of the trade balance-to-output ratio is to introduce financial imperfections as in the sovereign debt literature. T. Yun (SNU) Macroeconomics 1/07/ / 32

30 Key Features of GPU Model 1 Greenwood, Hercowitz and Huffman Preferences u t = v t {(C t τγ t 1 L ν t /ν) 1 σ 1}/(1 σ) 2 The social resource constraint is exp(a t )F (K t, Γ t N t ) = C t + I t + NX t + S t 3 Accumulation of international debt q t B t+1 = B t NX t 4 Capital evolves according to a linear accumulation technology with a quadratic adjustment costs: K t+1 = I t + (1 δ)k t + φ 2 (K t+1/k t exp(µ g )) 5 Elasticity of the interest rate to changes in indebtedness r t = r + ψ(exp(b t+1 /Γ t b) 1) + exp(µ t 1) 1 T. Yun (SNU) Macroeconomics 1/07/ / 32

31 Calibration of AG and GPU Models AG model GPU model β ψ σ 2 2 γ 0.36 N/A φ δ T. Yun (SNU) Macroeconomics 1/07/ / 32

32 Country Premium and Trade-Balance Adjustment 1 Trade deficits raise the net foreign debt position. 2 An increase in the net foreign debt position magnifies the country premium. 3 Larger country premia tend to increase domestic savings and discourage private investment, thus dampening the increase in the trade deficits. 4 As a result, a high elasticity of the interest rate with changes in the foreign debt position helps eliminate the near random-walk behavior of the trade balance-to-output ratio. T. Yun (SNU) Macroeconomics 1/07/ / 32