Question 3 A simple decentralized economy This question introduces a distinction between households and firms in a simple version of the decentralized economy. A typical household aims to maximize its expected discounted lifetime utility function given by: V. =>B U(G) where the period utility isU(G) = =1- and the discount factor = 140. 0 denotes the rate of time preference ando is the coefficient of relative risk aversion discussed in class. The household received exogenous incomeIt in each period which can be either consumed (t) or invested in financial capital assets d, which return the interest rate t. A representative firm hires capital ke at rental rater to produce output yt = F(kit). The produc- tion function is given by: F(kt) = kf while capital depreciates at rated. (a) Formulate the intertemporal budget constraint of the typical household and set up its in- tertemporal maximization problem. Derive the first order conditions and the Euler equation. (b) Set up the problem of the firm and derive the demand for production capital. (c) Define the competitive general equilibrium for this economy. Using an expression for the expected lifetime wealth of the household, in class we derived the following consumption function: Twt = r Ct = 1+r ( Ity)tti + rat 1=0 3where u depicts the wealth of the household. (d) Why has this solution been refereed to as the "permanent income hypothesis"? (e) Use the consumption function to illustrate the effects on current consumption of a perma- nent increase in income. (f) Compare your answer in (e) to a temporary change in income expected to occur in period t+ 2. (g) What policy implications might result as a consequence of these differences? Using the households' Euler equation, in class we showed that the growth rate of consumption on the optimal path is given by: de+1 = dct o(1+r) (h) Determine how the growth rate of consumption depends on the relative size of and 0. (i) Determine the equilibrium solution and explain why any other solutions will not be sustainable in the long run