BADM 604 Optimization Homework Prof. Silver

1. The Consolidated Coal Company (CCC) is located in a small town about 70 miles from Wheeling, W. Va. Except for a few clerical and retail jobs, virtually the only employment available is with CCC. The wage rate is $10 per hour. The cost of capital is $20 per unit. CCC's production function is give by:

Q = 20 K^.4 L^.4, where

Q = output of coal in tons per year,

K = quantity if capital used, and

L = number of man-hours of labor

The current price of coal is $25 per ton.

Find optimal coal output for CCC, as well as the amount of labor and capital employed, total revenue and total costs (labor and capital) and total profit.

2. One Saturday night I found myself alone and decided to spend the time watching TV, guzzling beer, and downing peanuts. I know my joint utility function for beer and nuts (assuming I don’t approach the point of negative returns) is given by:

U(B,N) = B^.5 N^.5, where

U(•) = utility, in utils,

B = number of beers, and

N = number of packets of nuts.

I also decided to budget myself to $10 for the evening. Beers are $1 per bottle and peanuts are 50 cents per packet.

Find the optimal number of beers and packets of nuts to buy for my “big night”, and my total utility.