all parts please

1. (20 points) [VC] A rm is using a Cobb-Douglas production function q = (K(\")) x (La\"0) in the creation of bottled soda. Let L] = the number of cases of bottled sodas, K = the number of machines, L = the number of workers hired, and a = %. (a) (2 points) [VC] Suppose that the rm only has access to K = 32 machines currently. If the current market price for a case is P = $22.50, the rental rate of captial r = $24, and wage rate is w; then, nd the optimal choice of L* the rm should hire (hint: choice will be a function of w)? (b) (3 points) [VC] In the town, the labour supply is inelastic in the short-run with L5 = 500. What would happen if the government set a minimum wage of w = {$10, $15, $20} (hint: think how many would be hired versus how many would want to work for each of the 3 wages and number may have a decimal point)? (c) (4 points) [VC] Suppose the market wage is w = $6 based on L5 = 500 which means the rm would produce q = 200 cases in the short-run. If the rm continues to produce q = 200 and is able to adjust K and L (with P = $22.50, w = $6, and r = $24), how many people would be hired and how much capital would be used (w: assume the labour supply LS will have time to adjust and is not xed). (d) (6 points) [VC] How do the short-run profits from part (c) compare with the outcome when the rm can adjust K and L? Use a sketch to explain why this would be the case (make sure to label your sketch). (e) (5 points) [VC] Suppose a new technology comes along allowing the rm to replace all their existing machines resulting in production of q = z X (K("')) x (Lu"0) where z > 1 or the rm can introduce a new method with the existing machinary making production q = (K(\")) x (zL)(1_\") for z > 1, i.e. there is a neutral teclmological change or a labour saving change in the method of production. Which technology would the rm prefer? (hint: if stuck try 2 = 8) How should the rm respond in the short-run? Is your answer the same for the long-run? Explain