QUIZS... its complte.. hel me

This problem analyzes the effect of future financial constraints on current investment decisions. Consider an economy with two dates, denoted by t = 1, 2. There are two goods: consumption and capital. There is a continuum of entrepreneurs and a continuum of consumers. All individuals have linear utility and consume only in period 2, so U = E[c2]. There is a fixed supply of capital k, initially owned by the consumers. Consumers are endowed with a large amount of consumption good in each period and they can store this good between dates, such that if they store one unit of the consumption good in t = 1 they get one unit of the good in t = 2. The entrepreneurs have access to a linear technology that produces A units of consumption good in period 2 per unit of capital they own. The consumers have access to a concave technology G( k), where k denotes the capital owned by consumers in period 2. Assume limk0G0 (k) = and G0 (k) = 0. The entrepreneurs enter period 1 with a given net worth N1 in terms of consumption goods. Assume agents can trade a risk-free bond b2 that pays an interest rate r. a) Argue that the gross rate of return on the risk-free bond is equal to 1 (i.e., the net return, r, is zero). b) Suppose that entrepreneurs face no borrowing constraints. State the optimization problems of an entrepreneur and a consumer. Show that the equilibrium capital price is q1 = A and the entrepreneurs buy k , where G0 (k k ) = A. c) Suppose that the entrepreneurs cannot borrow at all, so q1k2 N1. Find the equilibrium price and allocation, show that q1 A in equilibrium and that the expected utility of the entrepreneur is A q1 N1 (18) irrespective of whether the constraint q1k2 N1 binds or not. Show that q1 is increasing in N1 for N1 < Ak . Let's add one period prior to period 1, period 0. Now the economy has three dates t = 0, 1, 2. In t = 0 the entrepreneurs have initial net worth N0 (in consumption goods). Then they borrow b1 from the consumers and buy capital k1 at the price q0 (capital never depreciates). The technology in period 1 is the same as in period 2: consumers produce using the concave technology G(k) and entrepreneurs have a linear technology. However the productivity of the entrepreneurs at t = 1 is a random variable a distributed on [a, a] with E[a] = A. The productivity in t = 2 is fixed at A. 9 The entrepreneurs can only issue non-state contingent, risk-free bonds, so b1 has to satisfy (q1(a) + a)k1 b1 | {z } =N1(a) 0 for all a. (19) Note that now the asset price q1 is in general a function of the shock a and we denote it by q1(a). d) Assume that the entrepreneurs' initial net worth N0 is sufficiently large so that (19) is not binding, and that they are not constrained in period 1. Show that asset prices are q0 = 2A and q1(a) = A for all a, and that the capital owned by the entrepreneurs is constant at k . e) Now suppose that the entrepreneurs are fully constrained at date 1, as in part (c), but are completely unconstrained at date 0. Argue informally that now the equilibrium asset price q1 is weakly increasing in a. f) Write the entrepreneur's problem at date 0. Assuming the constraint (19) is not binding, derive the first order conditions for the entrepreneur's problem. Derive an expression for the equilibrium price q0 as a function of q1(a).