Question

Suppose price-taking firms have cost functions given by C(q) = 90 + 5q + 0.025q^2

1. What are the equations of marginal costs and average costs?

2. How much would the firm produce at prices of $9, $10, $11, and $12?

3. How much profit would the firm earn at prices of $9, $10, $11, and $12?

4. Graph the MC, AC. Indicate the profits at a price of $9 per unit.

5. What price would be charged in the perfect competitive equilibrium?

Answer 1

(a) C(q) = 90 + 5q + 0.025q^2

MC = dC(q) / dq

MC = 5 + 0.05q

AC = C(q) / q

AC =  (90 + 5q + 0.025q^2) / q

AC = (90 /q) + 5 + 0.025q

---------------------------------------------------------------------------------

(b) A perfeclty competitive firm produces at P = MC

P = 5 + 0.05q

Put P = 9 and solve for q

9 = 5 + 0.05q

0.05q = 9-5

0.05q = 4

q = (4 / 0.05)

q = 80

The firm will produce q=80 at a price of $9

--------------------------------------------------------------------------

P = 5 + 0.05q

Put P = 10 and solve for q

10 = 5 + 0.05q

0.05q = 10-5

0.05q = 5

q = (5/ 0.05)

q = 100

The firm will produce q=100 at a price of $10

--------------------------------------------------------------------------

P = 5 + 0.05q

Put P = 11 and solve for q

11 = 5 + 0.05q

0.05q = 11-5

0.05q = 6

q = (6 / 0.05)

q = 120

The firm will produce q=120 at a price of $11

--------------------------------------------------------------------------

P = 5 + 0.05q

Put P = 12 and solve for q

12 = 5 + 0.05q

0.05q = 12-5

0.05q = 7

q = (7 / 0.05)

q = 140

The firm will produce q=140 at a price of $12

--------------------------------------------------------------------------

(c) Profit = TR - TC

Profit = Pq - C(q)

At a price of $9, q is 80

Profit = ($9 * 80) - [90 + (5 * 80) + 0.025 (80)^2]

Profit = $720 - $650

Profit = $70

A firm will earn a profit of $70 at a price of $8

--------------------------------------------------------------------------------

Profit = Pq - C(q)

At a price of $10, q is 100

Profit = ($10 * 100) - [90 + (5 * 100) + 0.025 (100)^2]

Profit = $1000- $840

Profit = $160

A firm will earn a profit of $160 at a price of $10

--------------------------------------------------------------------------------

Profit = Pq - C(q)

At a price of $11, q is 120

Profit = ($11 * 120) - [90 + (5 * 120) + 0.025 (120)^2]

Profit = $1320- $1050

Profit = $270

A firm will earn a profit of $270 at a price of $11

--------------------------------------------------------------------------------

Profit = Pq - C(q)

At a price of $12, q is 140

Profit = ($12 * 140) - [90 + (5 * 140) + 0.025 (140)^2]

Profit = $1680- $1280

Profit = $400

A firm will earn a profit of $400 at a price of $12

--------------------------------------------------------------------------------

(d)

Q	AC	MC
0		5
10	14.25	5.50
20	10.00	6.00
30	8.75	6.50
40	8.25	7.00
50	8.05	7.50
60	8.00	8.00
70	8.04	8.50
80	8.13	9.00
90	8.25	9.50
100	8.40	10.00
110	8.57	10.50
120	8.75	11.00
130	8.94	11.50
140	9.14	12.00
150	9.35	12.50

Price & cost) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Quantity AC MC

At a price of $9, 80 units of output will be produced at an average cost of $8.13.

Hence, the black color outline rectangle is the profit at a price of $8.13.

---------------------------------------------------------------------------------

(e) A perfectly competitive firm produces at the following point in the long run.

i.e., P = MC = min.AC

MC intersects AC at its minimum point.

MC intersects AC at 60 units of output and corresponding AC and MC is $8.

P = MC = min. AC = $8

Hence, perfectly competitive firm will charge a price of $8 in the long run.