#2.) Suppose a representative household has access to the production technology:
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Question
#2.) Suppose a representative household has access to the production technology: f(k)--(k2 + 3). The household distributes production across consumption and saving each period so that ct +St f (kt). Savings each period turns into productive capital for use in production the following period: kt+,-St. Let's assume the household has a 50% marginal propensity to consume so that cr = 2/(h). (a) Find the household's capital transition equation. (b) Carefully graph the capital transition equation together with the 45-degree line. (c) Find the two steady-state levels of capital. (d) Linearize the transition equation around each of the steady-states and characterize their stability. (e) Characterize the lim kt as a function of the initial state koExplanation / Answer
The utility function is given as u(c,c)=ln(c)+*ln(c), where c is consumption today and c is consumption tomorrow, and is the time preference factor. The household budget constraint in the first period is: c + s = w - t and in the second period is:c' = (1+r)s -t'. Therefore consumer lifetime budget constraint will be c1+c2/(1+r)=w-t-s+s-t'/(1+r) = w - t - t'/(1+r) =M. Now we form the lagrangian function for the maximisation as L= ln(c)+*ln(c) + (M- c1 - c2/(1+r)). Then dL/dc1=1/c1 - =0 , dL/dc2= /c - /(1+r) =0 and dL/d=0, M=c1+c2/(1+r).
By combining these conditions we get c2/c1=(1+r). Thus M=c1+c2/(1+r), so the optimal consumptions will be c1*=M/(1+) and c2*= (1+r)M/(1+). By imposing t=0.1 and =0.99 we can get equilibrium wage, saving and optimal consumptions for two periods.
The firm profit function is Y=F(K,L)=A(K+L1-) subject to the constraint N=Kr+Lw, Then the optimaulity condition can be obtained by lagrangian function l =A(K+L1-)+(N-Kr-Lw), Similarly using the FOC we can have the optimal output with the optimal allocation of factors K and L.
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