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: Vt =_ B U(c) where the period utility is U(c) = ] and the discount 1 = 0 factor B = . 0 denotes the rate of time preference and o is the coefficient of relative risk aversion discussed in class. The household received exogenous income it in each period which can be either consumed (c) or invested in financial capital assets a, which return the interest rate r. A representative firm hires capital , at rental rate r, to produce output y = F(k,). The produc- tion function is given by: F(k) = ke while capital depreciates at rate o. (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