Question

A paper company dumps nondegradable waste into a river that flows by the firm's plant. The firm estimates its production function to be: Q = 6KW, where Q = annual paper production measured in pounds, K = machine hours of capital, and W = gallons of polluted water dumped into the river per year. The marginal products of capital and labor are given as follows: MPK = 6W MPW = 6K The firm currently faces no environmental regulation in dumping waste into the river. Without regulation, it costs the firm $7.50 per gallon dumped. The firm estimates a $30 per hour rental rate on capital. The operating budget for capital and waste water is $300,000 per year.

a. Determine the firm's optimal ratio of waste water to capital.

b. Given the firm's $300,000 budget, how much capital and waste water should the firm employ? How much output will the firm produce?

Answer 1

Solution :-

(a) :-

The firm estimates its production function to be: Q = 6KW

where Q = annual paper production measured in pounds,

K = machine hours of capital,

W= gallons of polluted water dumped into the river per year

The marginal products of capital and labor are given as follows: MPK = 6W, MPW = 6K

Dumping cost of water (d) = $7.50

Rental rate on Capital (r) = $30

Cost minimization requires that

MPW/MPK = d/r

6K/6W = 7.50/30

K/W = 0.25

K = 0.25W

W = 1/0.25K

W = 4K

[ W/K = 4 ]

The firm's optimal ratio of waste water to capital is => [ W/K = 4 ].

(b) :-

Given the firm's $300,000 budget,

So, cost line is given by

d x W + r x K = C

7.50W + 30K = 300000

Put W = 4K

7.50 x 4K + 30K = 300000

30K + 30K = 300000

60K = 300000

K = 300000/60

[ K = 5000 ]

So , the capital employed by the firm is K = 5000

Now,

W = 4K

Put K = 5000

W = 4 x 5000

[ W = 20000 ]

So, waste water employed by the firm is W = 20000.