Question

Triple C Ranch (TCR) grows corn.  

The market supply for corn is Q_s = -9 +4P Market demand is Q_d = 40 – 3P.  

TCR’s cost function is 26 + 2q + 0.1q², where q is the output of corn in ears.

1. Derive the average cost function, the marginal cost function, and the average fixed cost function, and graph them on a single graph.
2. What is the market price of corn?
3. What is TCR’s profit-maximizing output of corn? (Use calculus.) Calculate TCR’s profits – use Π = (P – AC)q

Answer 1

A. The cost function is given as 26+2q+.1q², average cost formula is= total cost/quantity. Plugging in, we get

Average cost=(26+2q+.1q²)/q=26/q+2+.1q.

Marginal cost=d(total cost)/dq=2+.2q.

Average fixed cost function= 26/q.

They have been plotted below in a single graph.

-30 20. 10 | 10 20 | 30 | | 40 70 50 0 ,

Blue line is the average fixed cost function. Green is the average cost function. Red is the marginal cost function.

b. Market price of corn will be where supply=demand. So,

-9+4P=40-3P

7P=49

P=7.

c. Profit maximizing output would be where MC=MR. As derived in part A, MC=2+.2q.

Revenue= Price*Quantity=((Q-40)/3)*Q=Q²/3-40Q/3

MR=2Q/3-40/3.

Equating MR and MC

2Q/3-40/3=2+.2Q

2Q-40=6+.6Q

1.4Q=46

Q=32.86.