Suppose Zippy's Banana Juice can produce according the following long-run production function.Q = 5 L_{2} + 20 K - 0.4 K_{2}where Q is gallons of juice per hour, L is labor hours, and K is capital-hours. Assume you can purchase a combined total of 6 units of K and L.The firm faces a market for inputs where PL = $25, PK = $40. Fixed costs are $50 per hour.
a. What are the profit-maximizing levels of K and L?
MPL/PL = MPK/PK10L/25 = (20 - 0.8K)/40Cross-multiply400L = 500 - 20KDivide by 2020L + K = 25 (Equation #1)Add constraint equationL + K = 6 (Equations #2)Solve two equations, two unknowns to getL = 1, K = 5
b. What is the profit-maximizing Q?Plug into the production functionQ = 5 L2 + 20 K - 0.4 K2Q = 5 (1)2 + 20 (5) - 0.4 (5)2Q = 5 + 100 - 10Q = 95Note: fixed costs do not affect the profit-maximizing level of L, K, or Q.
2. A publishing house uses two kinds of web content producers - writers (W) and poets (P). They produce web pages (Q) per day according to the production functionQ = 5W_{2} + 50P - 2P_{2}
Writers and poets are paid the same, $100 per day. The company sells the web pages for $20. Thecompany has a daily budget of $800 to hire writers and poets. How many writers and poets should be hired?