Question

Optimization Exercise

1. Fat Cattle Inc. cattle feeders uses the following cost function:

TC= 8,912 + 1.2 X^2 - .003 X^3

Find the production level (X) that minimizes costs.

2. The Smith Farm's profit is related in the following way to its output:

Profit= -240 + 8Q - 6Q^2

a. Find the firms output equals 6, what is the marginal profit?

b. What output will maximize the firms profit?

3. Jones Agriculture Inc. hires a consultant to estimate its profit function. The function is reported as:

Profit= -10 - 6Q + 5.5Q^2 - 2Q^3 + .25Q^4

a. The consultant says the firm should set Q equal to 1 to maximize profit. Is it true (dprofit/ dQ)=0 when Q=1?

b. Is profit a maximum when Q=1

C. The VP of Jones Agriculture says the profit is maximized at Q=2. Is this true?

Answer 1

Answer 1:  TC=8912+1.2x²-.003x³

For minimizing cost x should be such that dTC/dx=0.

dTC/dx=2.4x-.009x²=0 or x=266.667

To verify that the cost is minimum at this x, we calculate d²TC/d²x, if it is negative then the cost will be minimum at this x.

d²TC/d²x=-.018x=-ve.

Answer 2:

Profit(P)=-240+8Q-6Q²

a.

Marginal Profit =dP/dQ=8-12Q

at Q=6, Marginal Profit=8-72=-64

b.

Profit will be maximum when dP/dQ=0

8-12Q=0 or Q=8/12=0.667.

Answer 3: Profit=-10-6Q+5.5Q^2-2Q^3+.25Q^4

a. dprofit/dQ=-6+11Q-6Q^2+Q^3

at Q=1 dprofit/dQ=-6+11-6+1=0. It's true that when (dProfit/dQ)=0 when Q=1.

b.

Profit can be said to be maximum at Q=1 if d²Profit/d²Q=-ve

d²Profit/d²Q=11-12Q+3Q²

at Q=1, d²Profit/d²Q=11-12+3=2 that is not negative so Profit is not maximum at Q=1

c.

At Q=2, d²Profit/d²Q=11-12*2+3*2²=11-24+12=-1 is -ve. So, profit is maximized at Q=2.