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b) What is the solution for the path of the nominal interest rate following an increase in the inflation target? (Hint: one way to find this is by using the policy rule together with the solution you found in part (a)). c) First assume p = 0. Discuss how, and why, an increase in the inflation target affects nominal interest rates, the output gap and inflation. How would your results change if p = 1?\fQuestion 5 (0 points} This problem analyses the welfare effects of a \"capital injection\" in a model with nancial frictions. There are two periods, I] and 1. There are two types of agents, con- sumers and entrepreneurs. Both types have a linear utility function, cu+c1. Consumers have a large endowment of the consumption good in each period, and a unit endow- ment of labor in period 1 which they eel] inelasticelly in a competitive lahor marl-net at the prioe all. Entrepreneurs have an endowment of the consumption good on in period [1, and no initial capital. In period I} they can borrow b; and invest in. In period 1 they hire workers at the wage as and produce the consumption good according to the Cobb-Douglas production function y1=kiiiu Assume that k, completer depreciates after use in production. Moreover, the entrepreneurs face a borrowing constraint I5: 5 Mrs ' will}, where J. E (0,1) is a given scaiar. Assume that the consumers endowment is large enough so that the gross interest rate is always 1 in equilibrium. a) State the entrepreneurs' problem. Take the rst order condition with respect to 11. Show that the entrepreneurs always choose I, to maximize prots in period 1, that is, they choose 1 to solve to, = [1 o)icfi,'\" Show that this implies that prots are a linear function of the capital stock 1:1, that is, to 13151 = Hit-\"11331 \fe) Show that urban entrepreneurs are constrained, the equilibrium level of capital solves ' "a k\" = 1 Ayea Show that E1 and the entrepreneurs' prots are increasing in m]. Him: Remember that we limit attention to the ease in which li'w r: 1 5 Rhul}. f) Assume that the entrepreneurs are constrained. Suppose the government taxes a. lump-sum 1" to consumers in period 0, and transfers the receipts from the tax to the entrepreneurs. Derive an expression for the utility of oonsumers and entrepreneurs es a function of 1-. 3} Show that if u is suleiutly small, a small positive tax increases the utility of both consumers and entrepreneurs