Chapter 1 continued. Wouter J. Den Haan. University of Amsterdam. November 20, 2010

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1 Chapter 1 continued Wouter J. Den Haan University of Amsterdam November 20, 2010 Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

2 Environment of the competitive equilibrium Households: Firms: maximize discounted utility current-period utility depends on consumption and leisure own the capital stock, capital rented out each period at rate r t time endowment is equal to 1 which is split between leisure and working wage rate is equal to w t maximize NPV of pro ts competive input and output markets Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

3 Firm problem & FOC conditions α 1 a max θ k j k j,h j t ht j r t kt j w t ht j αθ t k j t h j t (1 α)θ t k j t h j t! α 1 = r t! α = w t Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

4 Firm size CRS =) rm size not determined Use representative rm without loss of generality k t = J j=1 kj t and h t = J j=1 hj t αθ t k t h t α 1 = r t (1 α)θ t k t h t α = w t The rm thinks of these equations as follows: given values for r t and w t choose k t and h t. Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

5 Individual problem max fc i t+τ,ht+τ i,k t+1+τg i E τ=0 β τ u(ct+τ, i 1 τ=0 s.t. ct+τ i + kt+1+τ i r tkt+τ i + w t ht+τ i + (1 kt+1+τ i 0 k t predetermined h i t+τ)ji t δ)k i t+τ Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

6 First-order conditions u(c i t, 1 h i t) c i t = λ i t λ i tw t = u(ci t, 1 ht) i ht i o λ i t + βe t nλ i t+1 [r t δ] = 0 c i t + k i t+1 = θ t k i t α h i t 1 α + (1 δ)k i t Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

7 Equations Equations for k t and h t from the rm problem taking r t and w t as given Equations for k i t,h i t, c i t, and λ i t from the household problem taking r t and w t as given 8i Equilibrium conditions k t = kt i i h t = ht i i Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

8 Simpli cation Suppose that all agents start with the same capital stock =) they also make the same choices k i t = k t, h i t = h t, c i t = c t, and λ i t = λ t Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

9 6 equations in 6 unknowns u(c t,1 h t ) c t = λ t λ t w t = u(c t,1 h t ) h t λ t + βe t fλ t+1 [r t δ]g = 0 c t + k t+1 = θ t (k t ) α (h t ) 1 α + (1 δ)k t r t = αθ t k t h t α w t = (1 1 α 1 α 1 = Ikt αθt Ih t = αθ kt t t h t t α)θ t k t h t α = (1 α)θt kt h t α unknowns: c t, k t+1, h t, λ t, r t, w t unknowns also could have been: c t, k t, h t, λ t, r t, w t Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

10 De nition of equilibrium De nition (competitive equilibrium): A competitive equilibrium consists of a consumption function, c(k, k, θ), a labor supply function, h(k, k, θ), a capital function, k +1 ( k, k, θ), an aggregate per capita consumption function, c( k, θ), aggregate per capita capital function, k +1 ( k, θ), aggregate per capita labor supply function, h( k, θ), wage function, w( k, θ), and rental rate, r( k, θ), Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

11 De nition of equilibrium (continued) solve the household s optimization problem, solve the rm s optimization problem, are consistent with each other, that is, c ( k, θ) = c( k, k, θ), k +1 ( k, θ) = k +1 ( k, k, θ), and h ( k, θ) = h( k, k, θ) 8 k, 8θ satisfy the aggregate budget constraint: c( k, θ) + k +1 ( k, θ) = θ k α h 1 α + (1 δ) k. Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

12 Representative agent and aggregate the equation above distinguish between the choices and state variables of the individual, c, k +1, and h, and the aggregate variables, c, k +1, and h. In a representative agent framework these are the same and in practice we only solve for c( k), k +1 ( k), and h( k). But, it is important to understand that the complete solution allows us to answer the question what will happen with (say) an individual agent s consumption when his individual capital stock increases Note that k does not change when the capital stock of one individual increases (because the e ect of each individual on the economy is nil) Instead of using aggregate variables you can also express them as per capita variables Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

13 Social Planner & CE What is a social planner? Maximizes utility but only worries about feasibility and not prices and transfers That is, the social planner problem for this CE is the same as our Robinson Crusoe economy If you substitute out the rental rate and the wage rate, then you see that the equations of the competitive equilibrium are the same as those of the social planner. Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

14 Many di erent agents Environment Result: Endowment economy Agents are ex ante the same but receive di erent realizations of the endowment Complete asset markets Economy can be represented with a representative agent economy Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

15 Optimization problem max c i,b+1 1,i,,bJ,i +1 s.t.c i + c i 1 γ 1 γ + βev(b1,i +1,, bj,i J j=1 q j b j,i +1 = y i + b j,i +1 > b < 0 J j=1 I (j )b j,i +1 ) Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

16 First-order conditions q j c i γ = β c j,i +1 γ prob(j) 8j Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

17 Explicit aggregation c i = C = βprob(j) 1/γ q j c j,i +1 8j or βprob(j) 1/γ C j +1 8j, q j γ q j (C ) γ = β C+1 j 8j γ q j (Y ) γ = β Y+1 j 8j Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18

18 Representative agent economy Any di erence? (C ) 1 γ max C,B+1 1,,B +1 J 1 γ + βev(b1 +1,, B J +1 ) s.t.c i + J j=1 q j B j J +1 = Y + I (j )B j j=1 B j +1 > b < 0 Wouter (University of Amsterdam) Chapter 1 continued November 20, / 18