The Madison Corporation, a monopolist, receives a report from a consulting firm

Question

The Madison Corporation, a monopolist, receives a report from a consulting firm concluding that the demand function for its product is Q=78 - 2P + 2Y + 0.9A where Q = # of units sold, P= price of products in dollars, Y= per capita income and A = firms advertising expenditure(in thousands of dollars). The firm's average variable cost function is AVC= 42 - 8Q + 1.5Q2 where AVC is average variable cost (in dollars),

a. can we determine the firms maginal cost curve?

b. can we determine the firms Marginal revenue curve?

c. If per capita income is $4,000 and advertising expenditure is $200,000 can we determine the price and output where marginal reveune eqauls marginal cost? if so what are they?

Explanation / Answer

(1)

AVC= 42 - 8Q + 1.5Q2

Since AVC = TVC / Q, TVC = AVC x Q = 42Q - 8Q2 + 1.5Q3

If Fixed costs be F, then

Total cost, TC = FC + TVC = F + 42Q - 8Q2 + 1.5Q3

So, Marginal cost, MC = dTC / dQ = 42 - 16Q + 4.5Q2

(b)

If Q = 78 - 2P + 2Y + 0.9A,

2P = 78 + 2Y + 0.9A - Q

P = 39 + Y + 0.45A - 0.5Q

Total revenue, TR = P x Q = Q x (39 + Y + 0.45A) - 0.5Q2

Marginal revenue, MR = dTR / dQ = 39 + Y + 0.45A - Q

(c) Y = 4000 and A = 200 [Since Y & A in thousands of dollars]

P = 39 + Y + 0.45A - 0.5Q

= 39 + 4 + (0.45 x 200) - 0.5Q

= 246 - 0.5Q

So, TR = P x Q = 246Q - 0.5Q2

MR = dTR / dQ = 246 - Q

MC = 42 - 16Q + 4.5Q2

Equating,

42 - 16Q + 4.5Q2 = 246 - Q

4.5Q2 - 15Q - 204 = 0

This is a quadratic equation whose roots are: Q = 8.6 or Q = -5.3. Dismissing the negative value, we get Q = 8.6

So:

(i) P = 246 - 0.5Q = 246 - 4.3 = 241.7

(ii) Output, Q = 8.6

NOTE: The quadratic equation has been solved using standard equation: ax2 + bx + c = 0 whose roots are:

-b ± (b2 – 4ac) / 2a

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