In: Economics
A film studio in Hollywood produces movies according to the function (yes, they can also produce fractions of movies... Think of half a movie as a B-movie or so.) q = F(K, L) = K0.5L 0.5 /100 (reads as K to the power of 0.5 times L to the power of 0.5 divided by 100). In the short run, capital (studios, gear) is fixed at a level of 100. It costs $4,000 to rent a unit of capital and $1,000 to hire a unit of labor (actors, stuntmen, camera crew etc.) The Hollywood studio is doing its planning for the next year and can choose capital and labor.
(a) Imagine that you come in as a new manager and discover that the current capital-labor ratio is K/L = 1. If you spend 10,000 additional (small fractions of) dollars on hiring more labor, how many additional (small) units of labor can you hire and how much more output can you produce? Answer the same for capital. If you had to stay on the same budget, would you hire or fire workers, in order to maximize output?
(b) Still assume that the current capital-labor ratio is K/L = 1. How many (small) units of capital can you save when you hire four additional (small) units of labor, holding output constant? How much money can you save when you do so?
Production function is q = F(K, L) = K^0.5L^0.5/100. In the short run, K is constant = 100. Here rental price r =
$4,000 and wage rate w = $1,000. The production
(a) Here capital-labor ratio is K/L = 1. Money available for hiring labor is 10000. With a wage rate of $1000, a total of
$10000/$1000 = 10 workers can be hired. With K/L = 1 or K = L, we have L = K = 10, the output is q =
10^0.5*10^0.5/100 = 0.10 movies. Now with only capital we rent $10000/$4000 = 2.5 units of capital. Output will be
q = 2.5^0.5*2.5^0.5/100 = 0.025 movies.
At the optimal mix of L and K, we will actually hire according to the budget equation
10000 = 1000*L + 4000*L...........(K is replaced by L since K/L = 1)
10000 = 5000L
L* = 2 workers so now we will fire 8 workers.
(b) The capital-labor ratio is K/L = 1. At the optimum, we have K = L = 2 and so output is q = 2^0.5 2^0.5/100 = 0.02. If you hire four additional (small) units of labor, holding output constant at 0.20, find the number of capital units
0.02 = (2+4)^0.5 * K^0.5/100
This gives K = 0.67. Hence you can save 1.33 units of capital The budget for this combination is 4000*1.33 + 1000*6 = 11320. So you will have to spend more money when you deviate from optimal input mix.