Question

Suppose that the XYZ Corp. (which is a profit-maximizing enterprise) produces “gadgets” according to the following production function: Q = 300K + 100KL + 2000L - L^2 , where Q is the number of gadgets per year, K is the amount of capital (machines) that are used, and L is the number of workers employed per year. Gadgets sell for $100 each.

a) If XYZ has 10 machines, and workers cost $100,000/year (including benefits and direct ancillary expenses), how many workers should XYZ hire?

b) If the workers cost $120,000, how many workers should XYZ hire?

c) If workers cost $100,000 but gadgets sell for $80 each, how many workers should XYZ hire?

d) If XYZ instead has 20 machines, how many workers should it hire when workers cost $100,000 and gadgets sell for $100?

Answer 1

a.

$Q=300K+100KL+2000L-L^2\\K=10\\Q=300(10)+100(10)L+20000L-L^2\\Q=3000+1000L+2000L-L^2\\MP_L=\frac{dQ}{dL}=1000+2000-2L\\=3000-2L$

At profit maximization, $MP_L=\frac{w}{p}$

$3000-2L=\frac{100000}{100}\\3000-2L=1000\\2L=2000\\L=1000$

1000 workers will be hired.

b.

Now w=120000

$3000-2L=\frac{120000}{100}\\3000-2L=1200\\L=900$

900 workers will be hired.

c.

Now w=100000 and p=80

$3000-2L=\frac{100000}{80}\\L=875$

875 workers will be hired.

d.

K=20

$Q=3000(20)+100(20)L+2000L-L^2\\Q=60000+2000L+2000L-L^2\\MP_L=\frac{dQ}{dL}=2000+2000-2L=4000-2L\\4000-2L=\frac{100000}{100}\\L=1500$

1500workers will be hired