Solve the following.,,,

Question 1 (total 15 points). A firm has a production function: f(K, L) = 50LK0-5 + 12 K2 - 13K3 a. What is its short-run production function if capital is fixed at K=4? b. What are the firm's marginal product of labour and average product of labour in the short run? c. Calculate the elasticity of output with respect to labour. Question 2 (total 16 points) . Do the following functions exhibit increasing, constant, or decreasing returns to scale? Explain your answers. a. The production function O = 1050-5105, where M is materials, K is capital and L is labour. b. q =L +0.5K c. q = 0.5LK'0.25 d. q=417 +4K Question 3 (12 points). Find the Marginal Rate of Technical Substitution for the following production functions: a. q= DKi b. q =L0.5 + K0.5 c. q =L+K Question 4 (Total 10 points). Suppose a firm's cost function is C = 2q' - 16q2 + 90q, average cost curve is described by the equation AC=2q- - 16q + 90. At what output level does the marginal cost curve cross the average cost curve? Question 5. (16 points, 8 points each) For the following, please answer "True" or "False" and explain why. a. When buying a piece of equipment, it is always best for the firm to pay cash instead of borrowing the funds since this renders the equipment less costly. b. University of Toronto is choosing a location for a new building. The university has campuses at Downtown, Scarborough and Mississauga. A large parcel of land would have to be purchased at Scarborough or Mississauga if the building were to -Page I- be built there. So it is cost-effective to locate the new building at downtown because the university already owns the land.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?Part II. Do the following questions with all necessary steps 1.Assume labor (L) is the only variable input used in the production process, a firm's production function is given by 0=74 +10 4- - L' where Q represents total product. Classify the production function in to the three stages of production. 2. Answer the following questions based on the following information Labor input (in hour) 0 2 3 4 S 6 7 8 9 10 Total Product( in kg) 0 40 120 220 312 390 455 515 565 565 563 . The firm uses capital as a fixed input and labor as a variable input. . The total cost of capital it uses is birr 100 and the price of labor is 40 per unit. Calculate:Question 2 (25 marks) In a one-period model economy, there are many consumers, many firms and a government. All consumers are identical. The representative consumer is en- dowed with h units of hours, and derives utility from both consumption c and leisure I. The utility function is w/(c, !) = Inc + all, where a > 0. Let w denotes real wage. He receives labor income w(h = 1), of which T proportion is paid as tax (proportional income tax). All firms are also identical to each other and the market is perfect competitive. The representative firm hires labor and produces consump- tion good with production technology given by Y = AN, where Y is output, A is productivity and N is labor demand. All the firms are owned by the consumers. The government can not issue bonds and runs a balanced budget. That is, the tax collected by the government is equal to its spending G." a) Define the competitive equilibrium for this economy. b) Solve for the competitive equilibrium - the equilibrium wage, consumption and leisure. c) Set up a social planner's problem and derive the efficient allocations. d) Is the competitive equilibrium socially optimal? Why or why not. Question 3 (25 marks) In a two-period model economy, there are a total N number of consumers, of which IN are of type A and N are of type B consumers. Consumers derive utility from current period consumption c and future period consumption c'. For all consumers, current and future consumptions are perfect substitutes. Type A consumers have MRSep = a, and type B consumers have MRScp = b, where a 0, Ny' - " Ny'-G' G' > 0 and a > 2 Ny-G. a) Solve for the competitive equilibrium - the equilibrium real interest rate, and the current and future consumption allocations for each individual consumer. b) Does the Ricardian Equivalence Theorem hold in the equilibrium? Explain why or why not.8. A firm uses inputs of labor, L, and capital, K, to produce its output, Q, according to the production function Q = f(K, L) = 91 1/3K1/3. The firm is a price-taker in the input markets. Labor is paid an hourly wage of w = 12, and the price of capital is r = 6. The firm sells its output at a price of P = 4 per unit. Maximize the profit function I(K, L) = Pf(K, L) - WL - rK to determine the optimum level of each input the firm should use