A firm has the production function f(k, l) = 2k√l. Let the price of capital be r = 1, the price of labor be w = 2, and the price of output be p.1
(a) Find the marginal products of capital and labor. Does the firm have constant returns to scale?
(b) What is the equation for the firms isoquant for output level equal to 8. Find three points on it.
(c) Suppose the quantity of capital is fixed at k = 2. Solve the firms short-run profit maximisation problem and find its optimal demand for labor and its profit as functions of the output price p. Show that the firms short-run profit maximizing output level is y∗(p) = 4p.
(d) Now suppose there are 5 firms in total in the market identical to the above firm.Therefore, the market supply function is qS(p) = 20p. The market demand function is qD(p) = 120 - 10p. Find the equilibrium price and quantity. How much is the consumer surplus? How much is the producer surplus? What is the price elasticity of demand at the equilibrium price and quantity? Is marginal revenue at the equilibrium quantity positive or negative?
(e) Suppose now a quantity subsidy, s = $1.5 per unit sold is given to the producers.Solve for the new equilibrium quantity and prices pS and pD after the subsidy is introduced. Do consumers pay more or less per unit than in (d)? Are more or less units sold? Explain why.
(f) Draw a graph to indicate the before- and after-subsidy equilibrium and how the consumer and producer surplus change (no need to compute the exact after-subsidy surpluses). What is the total amount of money needed to finance the subsidy? Is there a deadweight loss?