Lecture 15 - Dynamic Stochastic General …

1 Lecture 15. Dynamic Stochastic General equilibrium model Randall Romero Aguilar, PhD. I Semestre 2017. Last updated: July 3, 2017. Universidad de Costa Rica EC3201 - Teor a Macroecon mica 2. Table of contents 1. Introduction 2. Households 3. Firms 4. The competitive equilibrium 5. The central planning equilibrium 6. The steady state 7. IRIS. Introduction Dynamic Stochastic General equilibrium (DSGE) models DSGE models have become the fundamental tool in current macroeconomic analysis They are in common use in academia and in central banks. Useful to analyze how economic agents respond to changes in their environment, in a Dynamic General equilibrium micro-founded theoretical setting in which all endogenous variables are determined simultaneously.

2 Static models and partial equilibrium models have limited value to study how the economy responds to a particular shock. 1. DSGE: microfoundations + rational expectations Modern macro analysis is increasingly concerned with the construction, calibration and/or estimation, and simulation of DSGE models. DSGE models start from micro-foundations, taking special consideration of the rational expectation forward-looking economic behavior of agents. 2. Households General assumptions about consumers There is a representative agent. Who is an optimizer: she maximizes a given objective function. She lives forever: in nite horizon Her happiness depends on consumption C and leisure O.

3 The maximization of her objective function is subject to a resource restriction: the budget constraint. 3. Instant utility The instant utility function is u(C, O). She prefers more consumption and more leisure to less: uC > 0 uO > 0. Higher consumption (and leisure) implies greater utility but at a decreasing rate: uCC < 0 uOO < 0. 4. Expected utility function The consumer's happiness depends on the entire path of consumption and leisure that she expects to enjoy: U (C0 , C1 , .. , C , O0 , O1 , .. , O ). She's impatient: she discounts future utility by . Her utility is time separable. Therefore, her expected utility is .. E0 t u(Ct , Ot ). t=0. 5. Resource ownership To de ne a budget constraint we must introduce property rights.

4 Here,we assume that the consumer is the owner of production factors: capital K and L labor. L comes from the available endowment of time, which we normalize to 1. Because time cannot be accumulated, labor decisions will be static. K is accumulated through investment, which in turn depends on savings. Consumer also owns the rm. 6. The budget constraint Household income comes from renting both productive factors to the production sector, at given rental prices. Household can do two things with these earnings: expend it in consumption or save it. Then, the budget constraint is Pt (Ct + St ) Wt Lt + Rt Kt + t where Pt = price of consumption good St = savings Rt = user cost of capital Wt = wage t = rm's pro ts (= dividends).

5 Since there is no money, we normalize Pt = 1 t. 7. Resource constraints Since time is spent either working or in leisure: Ot + Lt = 1 t Given this constraint, in what follows we write the instant utility function as: u(C, 1 L). Because capital deteriorates over time, its accumulation is subject to depreciation rate : Kt+1 = (1 )Kt + It 8. The nancial sector To keep things simple, we assume that there is a competitive sector that transforms savings directly into investment without any cost. Thus S t = It Combining this assumption with the budget constraint and the capital accumulation equation, the consumer is constraint by Ct Wt Lt + Rt Kt + t St Wt Lt + Rt Kt + t It Wt Lt + Rt Kt + t + (1 )Kt Kt+1.

6 Wt Lt + t + (1 + Rt )Kt Kt+1. 9. The consumer problem The consumer problem is to maximize her lifetime utility .. E0 t u(Ct , 1 Lt ). t=0. subject to the budget constraint*. Ct = Wt Lt + t + (1 + Rt )Kt Kt+1 t = 0, 1, .. where K0 is predetermined. *. We impose equality because uC > 0. 10. The consumer problem: Dynamic programming The consumer problem is recursive, so we can represent it by a Bellman equation. Current capital is the state variable, next capital and labor are the policy variables. Then we write { }. V (K) = max . u(C, 1 L) + E V (K ). K ,L. subject to the budget constraint C = W L + + (1 + R )K K . 11. The consumer problem: solution The FOCs are: uO = W uC (wrt labor).

7 UC = E V (K ) (wrt capital). The envelope condition is V (K) = (1 + R ) uC. Therefore, the Euler equation is [ ]. uC = E (1 + R )uC . 12. Consumer optimization: In summary For the numerical solution of the model , we assume that u (Ct , 1 Lt ) = ln Ct + (1 ) ln(1 Lt ). Therefore, the solution of the consumer problem requires [ ]. Ct 1 = E (1 + Rt+1 ). Ct+1.. Ct = W (1 Lt ). 1 . Kt+1 = (1 )Kt + It 13. Firms The rms Firms produce goods and services the households will consume of save. To do this, they transform capital K and labor L into nal output. They rent these factors from households. 14. Production function Technology is described by the aggregate production function Yt = At F (Kt , Lt ).

8 Where Yt is aggregate output and At is total factor productivity (TFP). Production increases with inputs . FK > 0 FL > 0. but marginal productivity of each factor is decreasing : FKK < 0 FLL < 0. 15. Production function (cont'n). We assume that Production has constant returns to scale: At F ( Kt , Lt ) = Yt Both factors are indispensable for production At F (0, Lt ) = 0 At F (Kt , 0) = 0. Production satis es the Inada conditions lim FK = lim FL = . K 0 L 0. lim FK = 0 lim FL = 0. K L . 16. The rm's problem: static optimization Firms maximize pro ts, subject to the technological constraint. max t = Yt Wt Lt Rt Kt Kt ,Lt Yt = At F (Kt , Lt ). or simply max At F (Kt , Lt ) Wt Lt Rt Kt Kt ,Lt 17.

9 The rm's problem: solution The FOCs are: Wt = At FL (Kt , Lt ) (wrt labor). Rt = At FK (Kt , Lt ) (wrt capital). that is, the relative price of productive factors equals their marginal productivity. 18. Side note: Euler's theorem Let f (x) be a C 1 homogeneous function of degree k on Rn+ . Then, for all x, f f f x1 (x) + x2 (x) + + xn (x) = kf (x). x1 x2 xn 19. The rm's pro ts Since F is homogeneous of degree one (constant returns to scale), Euler's theorem implies [At FK (Kt , Lt )] Kt + [At FL (Kt , Lt )] Lt = Yt Substitute FOCs from rms problem: Rt Kt + Wt Lt = Yt and therefore optimal pro ts will equal zero: t = Yt Rt Kt Wt Lt = 0. 20. The total factor productivity The TFP At follows a rst-order autorregresive process: ln At = (1 ) ln A + ln At 1 + t where the productivity shock t is a Gaussian white noise process: t N (0, 2 ).

10 This assumption led to the birth of the Real Business Cycle (RBC) literature. 21. The total factor productivity (cont'n). The TFP process can also be written ( ). ln At ln A = ln At 1 ln A + t In equilibrium , At = A.. Productivity shocks cause persistent deviations in productivity from its equilibrium value: ( ). ln At+s ln A . = s 1 > 0. t as long as > 0. Although persistent, the effect of a shock is not permanent ( ). ln At+s ln A . lim = s 1 = 0. s t 22. Firm optimization: In summary For the numerical solution of the model , we assume that At F (Kt , Lt ) = At Kt L1 . t and A = 1. Therefore, the solution of the rm problem requires ( ) . Kt Yt Wt = (1 )At = (1 ). Lt Lt ( )1.