Suppose Zippy's Banana Juice can produce according the following long-run production function.
Q = 5 L₂ + 20 K - 0.4 K₂
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/PK
10L/25 = (20 - 0.8K)/40
Cross-multiply
400L = 500 - 20K
Divide by 20
20L + K = 25 (Equation #1)
Add constraint equation
L + K = 6 (Equations #2)
Solve two equations, two unknowns to get
L = 1, K = 5

b. What is the profit-maximizing Q?
Plug into the production function
Q = 5 L2 + 20 K - 0.4 K2
Q = 5 (1)2 + 20 (5) - 0.4 (5)2
Q = 5 + 100 - 10
Q = 95
Note: 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 function
Q = 5W₂ + 50P - 2P₂

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?