Problem set 4. The decentralized economy and log-linearization. Markus Roth. Chair for Macroeconomics Johannes Gutenberg Universität Mainz

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1 Problem set 4 The decentralized economy and log-linearization Markus Roth Chair for Macroeconomics Johannes Gutenberg Universität Mainz January 14, 2010 Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

2 Contents 1 Problem 1(Utility maximization and budget constraints) 2 Problem 2(Consumption smoothing, canceled) 3 Problem 3(Linearization) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

3 Problem 1(Utility maximization and budget constraints) Contents 1 Problem 1(Utility maximization and budget constraints) 2 Problem 2(Consumption smoothing, canceled) 3 Problem 3(Linearization) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

4 Problem 1(Utility maximization and budget constraints) The problem Theproblem is tomaximize V t = subject to three different budget constraints. β s U(c t+s ). (1) s=0 Weknowfrom ourpreviousanalysis thatthethreebudget constraints are equivalent. Equivalent means that we can transform the period budget constraint into the lifetime budget constraint and vice versa. Hence, we expect that the solution(the consumption Euler equation) should be identical. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

5 Problem 1(Utility maximization and budget constraints) Budget constraint 1 The first budget constraint we consider is a t+1 = (1+r)(a t +x t c t ). (2) Current assets and income that are not consumed is invested. The Lagrangian is L = β s U(c t+s )+λ t+s [(1+r)(a t+s +x t+s c t+s ) a t+s+1 ]. s=0 The first order conditions are L = β s U (c t+s ) λ t+s (1+r) =! 0 c t+s (I) L! = λ t+s+1 (1+r) λ t+s = 0 a t+s+1 (II) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

6 Problem 1(Utility maximization and budget constraints) Solution 1 Thesolutionis then or equivalently U (c t+s ) = (1+r)βU (c t+s+1 ), (3) U (c t+s ) βu (c t+s+1 ) = 1+r, where the marginal rate of substitution between consumption in periodtandconsumptiont+1 equalsthemarginal rate of transformation of consumption between both periods. This isthestandardeulerequationfor ageneralperiodutility function U( ). Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

7 Problem 1(Utility maximization and budget constraints) Budget constraint 2 The second budget constraint we consider is a t +c t = x t +ra t 1 or a t +c t = x t +(1+r)a t 1 (4) wherethedatingconventionis thata t denotestheendof period stockofassetsand c t and x t are consumptionandincome during period t. The Lagrangian is L = β s U(c t+s )+λ t+s [x t+s +(1+r)a t+s 1 c t+s a t+s ]. s=0 The first order conditions are L = β s U! (c t+s ) λ t+s = 0 c t+s (I) L! = λ t+s+1 (1+r) λ t+s = 0 a t+s (II) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

8 Problem 1(Utility maximization and budget constraints) Solution 2 Thesolutiontothisproblem is again theeulerequationfor the optimal intertemporal consumption decision U (c t+s ) = (1+r)βU (c t+s+1 ). (3) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

9 Problem 1(Utility maximization and budget constraints) Budget constraint 3 The last budget constraint we consider is ( ) 1 s ( ) 1 s c t+s = x t+s+(1+r)a t, (5) 1+r 1+r s=0 s=0 TheLagrangian tothisproblem is L = β s U(c t+s )+λ s=0 { s=0 [( ) } 1 s (x t+s c t+s)]+(1+r)a t 1+r Notethat in thiscase wehave onlyoneconstraint. Thus, there is only one Lagrangian multiplier λ. The first order conditions are ( L 1 = β s U (c t+s ) λ c t+s L c t+s+1 = β s+1 U (c t+s+1 ) λ ) s! = 0 (I) 1+r ) s+1! = 0 (II) ( 1 1+r Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23.

10 Problem 1(Utility maximization and budget constraints) Solution 3 Asalready expected,this leadsagain tothesame Eulerequation U (c t+s ) = (1+r)βU (c t+s+1 ), which weusually writefor periodt U (c t ) = (1+r)βU (c t+1 ). Wehaveshownthatexpressingthehousehold sproblemin anyof these three different alternative ways produces the same Euler equation. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

11 Problem 2(Consumption smoothing, canceled) Contents 1 Problem 1(Utility maximization and budget constraints) 2 Problem 2(Consumption smoothing, canceled) 3 Problem 3(Linearization) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

12 Problem 3(Linearization) Contents 1 Problem 1(Utility maximization and budget constraints) 2 Problem 2(Consumption smoothing, canceled) 3 Problem 3(Linearization) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

13 Problem 3(Linearization) The model Consider a nonlinear difference equation of the form x t+1 = f(x t ). (6) The linear approximation of this equation around the steady state x = x t = x t+1 is given by orequivalently usingx = f(x) x t+1 = f(x t ) f(x)+f (x)(x t x) x t+1 = f(x t ) x+f (x t x). This approximation issimply thetangentof thisfunction at the steady-state. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

14 Problem 3(Linearization) x t x t+1 Approx x t Figure: Approximationx t+1 = x t. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

15 Problem 3(Linearization) The approximation error Of course, the approximation error depends on the specific function you look at. Ingeneralyouwill have agoodapproximation whenx t is closeto its steady-state value x. Inour economicmodelsweusually assumethathis is thecase. Notethat theapproximation techniqueweapply hereis basedon a first order Taylor series expansion. Ingeneralitis also possibletocomputehigherorder approximations but we focus on the simplest case here. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

16 Problem 3(Linearization) Rewrite the function In order to log-linearize the difference equation we have to rewrite it in terms of log deviations around the steady-state. Thereforewedefineanewvariable ˆx t ln ( xt x ). Notethat wecan expresseveryvariable as x t = xeˆx t. Inthesteady-statethis newlydefinedvariable has tobezero ( x ˆx = ln = ln(1) = 0. x) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

17 Problem 3(Linearization) Rewriting the equation Usingthenewvariable wecan writethefunction as ( ) xeˆx t+1 = f xeˆx t. Nowwecan linearize this expressionas wedidabove. However,wenow use ˆx t insteadof x t andlinearize theleft hand side(lhs)and therighthandside(rhs) oftheequation separately LHS xeˆx +xeˆx (ˆx t+1 ˆx) = x+xˆx t+1 RHS f(xeˆx )+f (xeˆx )xeˆx (ˆx t ˆx) = f(x)+f (x)xˆx t. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

18 Problem 3(Linearization) Equating both sides Equating the left hand side approximation and the right hand side approximation yields x+xˆx t+1 = f(x)+f (x)xˆx t xˆx t+1 = f (x)ˆx t ˆx t+1 = f (x)ˆx t. Next, consider a Cobb-Douglas production function Y t = F(K t,l t ) = K α t L1 α t. (7) Wewriteit as Yeŷt = K α L α e αˆk t e (1 α)ˆl t. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

19 Problem 3(Linearization) The production function Theleft handsideis approximatedby LHS Y+Yŷ t. Therighthand sideisapproximated by RHS K α L 1 α +K α L 1 α [αˆk t +(1 α)ˆl t ] (8) Note that some intermediate steps are omitted. Equating both sides yields Y+Yŷ t = K α L 1 α +K α L 1 α [αˆk t +(1 α)ˆl t ] Yŷ t = K α L 1 α [αˆk t +(1 α)ˆl t ] ŷ t = αˆk t +(1 α)ˆl t. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

20 Problem 3(Linearization) The production function(general) In order to interpret the coefficients consider a general production function Y t = F(K t,l t ). Itis rewrittento ) (Keˆk Yeŷt = F t,leˆl t Theleft handsideis approximatedby LHS Y+Yŷ t. Therighthand sideisapproximated by RHS F(K,L)+F K (K,L)Kˆk t +F L (K,L)Lˆl t. (9) Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

21 Problem 3(Linearization) The production function(general) 2 Equating both approximations yields Y+Yŷ t = F(K,L)+F K (K,L)Kˆk t +F L (K,L)Lˆl t Yŷ t = F K (K,L)Kˆk t +F L (K,L)Lˆl t ŷ t = F KK Y ˆk t + F LAL Y ˆl t. From this representation it becomes clear that the coefficient on capital equals the elasticity of output with respect to capital. Maybe youare more familiar with F K K Y = Y K KY = ε YK. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

22 Problem 3(Linearization) Log-linearization and elasticities In general, when you log-linearize a function say y = f(x,z). The log-linearized function is given by ŷ t = ε yxˆx t + ε yz ẑ t. where ε yx and ε yz are therespectiveelasticities. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23

23 References References Wickens, M.(2008). Macroeconomic Theory: A Dynamic General Equilibrium Approach. Princeton University Press. Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, / 23