Question

Suppose Dan’s cost of making pizzas is C(Q) = 4Q + (Q2/40), and his marginal cost is MC = 4 + (Q/20). Dan is a price taker. (a) What is Dan’s supply function? (b) What is Dan’s supply function if he has an avoidable fixed cost of $10? [HINT: Recall that Dan will not supply anything unless P > min AC(Q). So, as a first step, you need to find AC(Q) from C(Q). In part (a), finding min AC(Q) is easy and you should be able to do so just by looking at the formula for AC (Q). For part (b), you can find the minimum of AC by using the fact that AC(Q) = MC(Q) at the minimum point of AC.]

Answer 1

(a) Dan’s supply function S(P) can be stated as follows:

S(P)= 0 If P<4.

And S(P) = 20P- 80 If P≥4

(b) Dan’s supply function S(P) can be stated as follows:

S(P)= 0 If P<5.

And S(P) = 20P- 80 If P≥5.

Step-by-step explanation:

Note that the equations given in the question can be correctly stated as follows:

C(Q) = 4Q + (Q^2/40) .................. (1)

MC = 4 + (Q/20) ............................ (2)

Therefore, we can now proceed as follows:

(a) What is Dan’s supply function?

The upward portion of the MC curve is the supply function of Dan.

Equating equation (2) to P, we have:

P = 4+ (Q/20)

P- 4 = Q/20

Q = 20P -80

The shutdown rule is that P > AVCmin

AVC = C(Q) / Q .................. (3)

Substituting equation (1) into (3), we have:

AVC = ( 4Q + Q^2/40)/ Q

AVC = 4 + (Q/40) ............... (4)

Since MC cuts the AVC at its minimum, equations (2) and (4) are then equated to solve Q which is the output level at which AVC is minimum as follows:

4 + (Q/20) = 4 + (Q/40)

(Q/20) - (Q/40) = 4 - 4

(Q/20) - (Q/40) = 0

Q = 0

Substituting Q = 0 into equation (4), we have:

AVCmin = 4+ (0/40)

AVCmin = 4

This implies that Dan will produce at any price ≥ $4.

Therefore, Dan’s supply function S(P) can be stated as follows:

S(P)= 0 If P<4.

And S(P) = 20P- 80 If P≥ 4.

(b) What is Dan’s supply function if he has an avoidable fixed cost of $10?

Since there is now a fixed cost, equation (1) becomes:

C(Q) = 4Q + (Q^2/40) + 10 ................. (5)

And the average cost (AC) will be as follows:

AC = (4Q + (Q2/40) + 10)/Q

AC = 4 + (Q/40) + (10/Q) .................... (6)

Since AC = MC when AC at its minimum, equations (2) and (6) are therefore equated to solve for Q as follows:

4 + (Q/40) + (10/Q) = 4 + (Q/20)

(Q/40) + (10/Q) = (Q/20)

Q = 20

Divide through by Q, we have:

(1/40) + (10/Q^2) = (1/20)

10/Q^2 = (1/20) - (1/40)

10/Q^2 = 0.05 - 0.025

10/Q^2 = 0.025

Q^2 = 10 / 0.025

Q^2 = 400

Q =
`√(400)`

Q = 20

Substituting Q = 20 into equation (6), we have:

AC = 4 + (20/40) + (10/20)

AC = $5

This implies that Dan will produce at any price ≥ $5.

Therefore, Dan’s supply function S(P) can be stated as follows:

S(P)= 0 If P<5.

And S(P) = 20P- 80 If P≥ 5