Replication of "Closing small open economy models"

Macroeconomics II Essay 3E106 Diana Aguiar – 110421007 João Ricardo M. G. Costa Filho – 110421002 2011/2012 Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 INTRODUCTION In the paper “Closing small open economy models” Schmitt-Grohé and Martín Uribe analyzed five different model specifications and concluded that the transition dynamics is the same regarding impulse-response function and second order moments. Among the models presented there was one small open economy with incomplete asset markets that assumes a debt-elastic interest-rate meaning that domestic agents face an interest rate that increases with the country’s net foreign debt. The aim of this paper is to derive and replicate the quantitative analysis done in the original work. After solving the model and defining the system of equations, the effects of an exogenous shock on technology on the main variables of the system – namely output, consumption, investment, hours of work, trade balance, interest rate and debt level – during the deviation of the original steady state until the convergence to the equilibrium are analyzed. The main conclusion is that, at the moment of the shock, output rises, consumption also increases, but with a lower magnitude; investment augments but eventually must be bellow the equilibrium value; hours of work increase less than proportionally to the increase in output. Differently from the previous variables, the trade balance experiences a negative initial shock. Finally, foreign debt and interest rate increase. This paper is organized as follows. The next section provides the assumptions of the model, the states of the optimization problem and the resulting system of equations that determines the dynamics of the model. The second part presents the steady state of the model. The third part focuses on the transition dynamics between two steady states, deriving loglinearized deviations from the equilibrium. The last part presents the simulation of the transition dynamics of the main variables given an exogenous shock on total factor of productivity. Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 1. THE MODEL In a small-open-economy set up, households behave in a rational way, maximizing its present-valued expected lifetime utility: ( ,ℎ ) ∞ With = 1 and = . The instantaneous utility has the following functional form: − ( ,ℎ ) = ℎ 1− −1 As is usual in this kind of model, households have a positive but decreasing marginal utility of consumption and an increasing marginal disutility of labor, represented in the model by the hours of work. The domestic agents can use their own resources as well as foreign capital. Therefore, the foreign debt dynamics is given by: where ( − = (1 + )= ! " ( − ) − + + + ( − ) )" represents the costs of accumulating capital. In the equation above, the current debt is a function of the last period´s stock of debt – plus the one-period of interests – less the savings made for paying part of the debt (i.e., aggregate income less the amount spent in consumption and investment and the last contemplating capital accumulation costs). The interest rate is a function of the equilibrium interest rate and the country´s foreign indebtedness level, as follows: = + #$ % & = + '" (( )+ * ), − 1) The production technology assumes Hicks neutrality regarding the technological progress and has the standard Cobb-Douglas functional form: = - .( , ℎ ) Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 with .( , ℎ ) = / ℎ / The total factor of productivity is assumed to be exogenous and follows an AR(1) process: where 3 01- = 201- + 3 is a white noise. The capital accumulation is given by the current stock of capital net of depreciation plus the flow of investments: + (1 − 4) = Under the presented assumptions the optimization problem is given by: max ( ,ℎ ) ∞ subject to = (1 + ) − + + =1 = = + '" (( )*+ ), − 1) = - .( , ℎ ) = + (1 − 4) given 01- = 201- + 3 - + ( − ) Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 One more restriction should be imposed to assure a Non-Ponzi dynamics of the system. The transversality condition is given by: lim : ∏:= :→∞ (1 + = ) ≤0 which implies that the present value of the debt should be less or equal to zero, i.e., no remaining debt in the limit. Setting up the Lagrangian: @= ( ,ℎ ) + A ∞ ( ,ℎ ) + A ( ⟺@= + ( ( ,ℎ − ( − − (1 + ∞ − (1 + " )+A − ) The first order conditions with respect to =0⟺A − ) F( + − + - .( , ℎ ) − − (1 + ) ( , ,ℎ , ,ℎ ) − A = 0 ⟺ − +- − − ( − ( .( − ,ℎ − )− ) ⟺ ) are: (1 + ) = 0 ⟺ A = A =0⟺ ) A − $1 + + '" (( )+ * ℎ ), =A − 1)& (24) (25) Equation (24) can be seen as follows. A household will postpone consumption up to the point where the decrease in present utility is equal to the present value of the expected increase in the future utility from the marginal consumption provided by the savings plus its remuneration. Equation (25) provides the relation of marginal utility of consumption and the Lagrangian mulplier (shadow price, i.e., by relaxing the constraint in one unit what changes on the objective function maximization) ℎ =0⟺ H( , ℎ ) + A - .( , ℎ ) = 0 ⟺ ℎ = A - (1 − I) / ℎ / ⟺ − ℎ Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 ⟺ ℎ = (1 − I) (26) Equation (26) presents a standard result. Households will increase hours of work up until its marginal disutility it gets from working is equal to the marginal utility it has from the income generated by working. In the case of perfect competition in the labor market, the wage (labor income) is equal to the marginal productivity of labor, as in the foresaid equation. =0⟺ ⟺ ⟺ A 1 + L( A 1 + L( = A 1+ − − A ′ ( − .K ( - ) = A ) = A ) - ) + (1 − 4) + ,ℎ ℎ / I / ′ ( − " + (1 − 4) + L( + 1 − 4 + L( " − ) " ) ⟺ − ) (27) The equation above shows how the allocation between consumption and investment evolves. Given the fact that capital is one input in the production function (and its marginal productivity is positive), an increase in investment implies, holding everything else constant, that future production increases, thus aggregate income augments and future consumption rises. This allocation occurs up to the point where the present value of the expected gain (weighted by capital´s share in the production function) is equal to the current loss, as usual. The interaction with the rest of the world is not only through debt, but also via international trade. Therefore the last relation to be found is the trade-balance-to-GDP ratio. Rearranging the terms of the debt equation yields: − − − ( − ) = (1 + = N1 + + '" $( )+OP = RS ⟺ 1 − ⟺1− − * ), − − − ) − 1&Q L ( 2 L ( 2 − − − − ⟺ − ⟺ )" = )" = RS RS − − ⟺ − − L ( 2 − (28) L ( 2 − )" − )" Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 The trade balance is given by the aggregate income less consumption, investment and investment costs. Notice that the trade balance already includes the service of the debt. From the equations derived, it is convenient to rewrite the system of equations that provides the dynamics of the model: =A = A 1 + L( − ) = = (1 + RS A / ℎ + (1 − 4) = $1 + + '" (( )*+ − ℎ =A ℎ = (1 − I) A ) =1− 01- (22) / − VI − + − ), (23) − 1)& (25) (26) + 1 − 4 + L( + L ( 2 = 201- + 3 = + '" $( )*+ ), (24) + ( − − 1& − )" " ) (30) − (29) )W (28) (27) (31) 2. STEADY STATE In this section, the steady state of the model is characterized. Since only equilibrium relations are involved, no time subscript is needed. The previous system of equations now becomes: =- / ℎ / = + (1 − 4) ⟺ = 4 (22) (23) Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 1 = $1 + + '" (( − 1)& ⟺ 1 = (1 + ) ⟺ ℎ = (1 − I) 1= − ⟺ ℎ = (1 − I) ZI ̅= 1−] = =A ℎ + 1 − 4[ + − − (25) / 1 1+ (24) (26) / (27) (28) ^ = RS 01- = 201- ⟺ - = 1 + '" (( − 1) ⟺ = (29) (30) = (31) Solving the System: Plugging equation (26) and (29) in (22): = / (1 − I) / ⟺ ( / /) = ( )/ (1 − I) / Substituting equation (24) in equation (27) and rearranging the terms: 1= 1 ZI 1+ + 1 − 4[ ⟺ +4 I = Replacing (27) in (26) yields: ( /) / I / =( ) V(1 − I) +4 / W⟺ I =( ) +4 / / / _(1 − I) N / Q( / /) ` After calculating the steady-state value of y only in terms of parameters, the other variables in steady state will be found only as function of y and the parameters: ℎ = (1 − I) / / Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 Hours of work in steady state depend on labor share in the production function (positively), income (also positively) and labor wage elasticity of labor supply1 (negatively). I +4 = The stock of capital in the steady state is positively related to the capital share in the production function but negatively related to the equilibrium interest rate and the depreciation rate. That means that, at the steady state, a greater interest rate would imply in a lower discount factor, thus consumers value the future less and current consumption would increase relatively to a steady state with lower interest rates. The depreciation acts in the same direction, since it increases the amount of investment to be made and therefore decreases consumption (relative to an equilibrium with a lower depreciation rate). =4 At the steady state, the investment made is only to replace the depreciated capital. = ̅ − − The consumption level of equilibrium is the steady state aggregate income less the equilibrium flow of investment and the interests paid on foreign debt. RS = 1 − + The log of trade-balance-to-GDP ratio is, therefore, the resulting amount of income after consuming and investing. A= − ℎ The shadow price in equilibrium is equal to the marginal utility of consumption. 1 From equation (26) ℎ = (1 − I) Therefore, ℎ = c P dOP ⇒ 3H,f = ∙ ⟺ℎ f dOP f ∙ f H = = ( /)a+ b+ . The RHS of the represents the real wage: ( /)a+ b+ =c. Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 3. TRANSITION DYNAMICS In order to analyze how the system responds to a shock on exogenous variables (namely a shock on productivity), the first step is to derive the log-linearized forms of the system of equations presented in section 1 and characterize deviations of the equilibrium. The log-linearization for each of the nine equations of the system is the following: = • - / / ℎ (1 − I) ℎ - + -I / / )h+ h )H+ H + -I ⟺ / / ℎ ℎ / )K+ K + -(1 − I) ⟺ i = -j + I k + (1 − I)ℎk • • + (1 − 4) = (1 − 4) k ⟺ A = ( + A A + ) k + • A P u = − A • ℎ P u − )p+ p ⟺ ℎ A ⟺ )K+lP K + '" $( )+ + '" * A A / )H+ H Aj = −ℎ ℎk = i ⟺ / ⟺ ℎ ⟺ )a+ a = )a+ a = )h+ h +I )K+ K = )m+ m + (1 − 4) ℎ ), & A )K+ K ⟺ k = 4n̂ + (1 − 4) k − '" A + A'" % − '" A = n̂ + + (23) ⟺ A = ⟺ A = )p )q p + A'" % ⟺ A p + = $ (1 + )&A +p +lP + Aj P )H+ H (26) + rs ), t ℎ =− A ̂ − ℎ ℎk = − A u Aj ℎ = (1 − I) (1 − I) i ⟺ ℎ ℎ (22) ⟺ + '" ))* A'" ̅ , + ⟺ ) / / = 4 n̂ + (1 − 4) k ⟺ k A A + -(1 − I) / %j (24) P u A ⟺ (25) = (1 − I) )a+ a ⟺ )F+ F −ℎ )H+ H (1 − I)ℎk = = Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 • A + A L( )= I A − a+lP K+lP ) ⟺ A + L( − ) A + AL + (1 − 4) I Ks a ⟺ A = −AL LA I p K − IA ) + IA N a )a+lP K Aj = L $ k − k 4 & Aj • • ̅ as F m a • − − i − !K a m + (29) + + + AL + LA )K+lP K a K " Q + AL( −k + + ( " + − ! "a+ !K a −2 + !K = 201-j + 3 a − ⟺ -j = '" % ⟺ ̂ = '" ̅ %j !K (30) (31) a − L + a K A A A NI + 1 − 4Q − − + = I LA &+ Notice that A=1 in steady state • " + ( − 01- = 201- − 01- + 3 ⟺ -j A ⟺ ̅ j = ̅ (1 + ) j − = 1 − a+ − a+ − F − − AL + L & + I $i ) + 01- 3 a (27) = (1 + RS F m a Ks A + (1 − 4) A a K )⟺ + + I + LA A ⟺ A = AL( )+ L$ k L( " NI + 1 − 4Q −k ̅ ̂ ) ⟺ RS + a K &+ " p K − − " − A NI + 1 − a K = (1 + ) + + vR − i R − j R =− a − − a ⟺ (28) + w = − F ̂ − m n̂ + ⟺ RS RS a a = (201- − 201-) + 201- − 01- + Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 4. SIMULATION After solving the optimization problem, deriving the equilibrium and characterizing the dynamics of the system, some simulations are provided using the same parameterization used by the authors. Using Matlab and Dynare, a temporary positive productivity shock has the following impacts on the system. The following graphs present the transition dynamics: Graph 1 – Transition dynamics y c 2 1.5 i 1.5 10 1 5 0.5 0 1 0.5 0 2 4 6 8 10 0 2 4 h 6 8 10 -5 2 4 tby 1.5 8 10 6 8 10 a 1 1.5 0.5 1 6 1 0 0.5 0 0.5 -0.5 2 4 6 8 10 -1 2 4 d 0.02 0 0 -2 -0.02 2 4 8 10 6 8 10 0 2 4 r 2 -4 6 6 8 10 -0.04 2 4 Source: authors´ elaboration The initial response of output is to rise and, since the shock remain only for one period, the deviation of the steady state tails off. A rise in income produces a rise in consumption but with a lower magnitude. Investment also increases initially but some disinvest must be done eventually for returning to a sustainable steady state level (otherwise the new stock of capital would be greater than the previous steady state, implying more investment to replace depreciated capital, diminishing steady state consumption and social welfare). The shock induces more production and thus more demand for labor. However, due to the effect of productivity, households increase hours of work less than the increase in output. Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 Differently from the previous variables, the trade balance experiences a negative initial shock. A rise in income implies more imports for a given level of exports, thus the trade balance becomes negative. For financing the current account deficit, more external debt is needed, and thus the financial account has a surplus and also debt increases. Reflecting the indebtedness level, the interest rate also rises. Since the stock of debt achieves during the transition dynamics lower levels than the steady state, the interest rate also achieves lowerthan-the-steady-state values. The productivity by construction follows an autoregressive process. Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 5. CODES %---------------------------------------------------------------% 1. Defining variables %---------------------------------------------------------------var y c i k h a tby d r lambda; varexo ea; parameters beta gamma delta phi alpha omega rho rbar siga dbar psi2 ybar hbar kbar ibar cbar tbbar lambdabar; %---------------------------------------------------------------% 2. Calibration %---------------------------------------------------------------beta=0.96; %discount factor gamma=2; %intertemporal elasticity of substitution delta=0.1; %rate of depreciation phi=0.028; %cost of capital adjustment alpha=0.32; %share of capital omega=1.455; %exponent of labor in utility function rho=0.42; %AR1 of tech process rbar=0.04; %(1/beta-1) is NOT 0.04. To make the calibration the same use rbar!!! siga=0.0129; dbar=0.7442; psi2=0.000742; ybar=(alpha/(rbar+delta))^(alpha/(1-alpha-(1-alpha)/omega))*(1-alpha)^(((1alpha)/omega)*1/(1-alpha-(1-alpha)/omega)); hbar=((1-alpha)*ybar)^(1/omega); kbar=ybar*alpha/(rbar+delta); ibar=delta*kbar; cbar=1.11695007562489;%ybar-ibar-rbar*dbar; tbbar=1-(cbar+ibar)/ybar; lambdabar=(cbar-(hbar^-omega)/omega); %---------------------------------------------------------------% 3. Model %---------------------------------------------------------------model(linear); y=a+alpha*k(-1)+(1-alpha)*h; %eq. 22 k=delta*i+(1-delta)*k(-1); %eq. 23 lambda=lambda(+1)+((psi2*dbar)/(1+rbar))*d; %eq. 24 cbar*c = h*hbar^omega-(1/gamma)*lambda*lambdabar^(-1/gamma); %eq. 25 omega*h=y; %eq. 26 lambda+phi*(k-k(-1))=beta*(alpha/kbar*ybar/kbar*y(+1)alpha/kbar*ybar/kbar*k+phi*(k(+1)-k))+beta*(alpha*ybar/kbar+1delta)*lambda(+1); eq. 27 tby=((cbar+ibar)/ybar)*y-(cbar/ybar)*c-(ibar/ybar)*i; %eq. 29 a=rho*a(-1)+ea; %eq. 30 rbar*r=psi2*dbar*d; %eq. 31 dbar*d=dbar*(1+rbar)*d(-1)+rbar*dbar*r(-1)+ybar*y-cbar*c-ibar*i; %eq. 28 end; Macroeconomics II – 3E106 Doctoral Program in Economics 2011/2012 %---------------------------------------------------------------shocks; %var ea = siga^2; var ea = 1; end; steady; stoch_simul(ar=1,irf=10) y c i k h a tby d r lambda; %---------------------------------------------------------------% 5. Some Results %---------------------------------------------------------------statistic1 = sqrt(diag(oo_.var(1:6,1:6)))./sqrt(oo_.var(1,1));