Please help me answer the following questions , thanks.,,,

Consider the two-period endowment economy with N consumers, and a government. Assume that the current and future consumption are perfect compliments. at is the preferences of each consumer are represented by the following Leontief utility function: U (c, c0 ) = min {ac, c0 } , where a > 0.

As Williamson (2014) puts it, perfect complementarity is an extreme case of a desire for consumption smoothing, in that the consumer never wants to deviate from having current and future consumption in fixed proportions. Suppose the current and future government expenditures are given exogenously by G and G0 , and current and future endowments of the consumers are given by y1, y2, . . . , yN and y 0 1 , y0 2 , . . . , y0 N .

(a) Define a competitive equilibrium of this economy in which the government levies the same lump-sum tax of t and t' on all the consumers in the current and future periods, respectively, and issues B units of bonds in the current period.

(b) Derive consumer's present value budget constraint, and write down the consumer's problem analytically, subject to the lifetime budget constraint.

(c) Given t, t 0 , yi , y 0 i , and the real interest rate r, solve for the current and future consumption of the consumer i. (d) Find the aggregate private savings, Sp , in the economy, and use it to and the equilibrium interest rate in terms of the exogenous variables.

Firm Optimization 6. (8 points) Suppose a firm uses capital (K), labor (L), and energy (E) to produce output (O). In the short-run, the firm cannot adjust its capital stock. The price of capital is r, price of labor is w, and the price of energy is e. The price of final output is p. In the short-run profit maximization problem, which of the variables K, L, E, O, p, r, w, e are exogenous? 7. (10 points) Assume a firm has the production function is Q = VKL . To maximize the firm's profit, it satisfies the condition w = pMPL. In the short-run, the capital stock is 9. If the wage rate is $6 and the price p is $18, how much labor does the firm demand? Markets Consider the following cost function: c(q) =16+15q+4q' 8. (8 points) Find the firm's individual supply function. 9. (8 points) Find the firm's minimum efficient scale. 10. (8 points) Suppose there are 20 firms with this cost function. What is the market supply function? II. (8 points) What is the long-run supply function of the market?I. A nn's production function is given by is q = KL 0.8K2 (3.2!.2 where q is the quantity of output produced, I. is the amount of labor input used, and K is the amount of capital input used. Suppose the amount of capital is fixed at 10 units of capital in the short run. I. 2. Find this fmn's short run (a) total product of labor function. (b) average product of labor function. and (c) marginal product of labor function. At what level of labor input usage does the marginal productivity of labor reach a maximum? How much output is produced at that point? Show and briey discuss how you arrived at your answer. Is the marginal product of labor ever negative for this production function? If so, when. If not why not. Show how you arrived at your answer. At what level of labor input usage does the average productivity of labor reach a maximum? How much output is pnoduced at that point? Show and briey discuss how you arrived at your answer. . At what level of labor input usage does the total productivity of labor reach a. maximum? How much output is produced at that point? Show and briey discuss how you arrived at your answer. What is the economic region of the short run production function you found in part I? Show and briey explain how you arrived at your answer. Calculate the output elasticity of labor when this rm uses the amount of labor at which its average product of labor is maximum. In a carefully labeled diagram, graphically illustrate this rm's total product of labor in the upper panel and the marginal product of labor and average product of labor in the lower panel using your results in parts 1 to 4