Question

6. Exercise 10.8

The Poster Bed Company believes that its industry can best be classified as monopolistically competitive. An analysis of the demand for its canopy bed has resulted in the following estimated demand function for the bed:

P=1,265−9QP=1,265−9Q

The cost analysis department has estimated the total cost function for the poster bed as

TC=Q33−15Q2+5Q+24,000TC=Q33−15Q2+5Q+24,000

Short-run profits are maximized when the level of output is

and the price is

.The total profit at this price-output level is

.

The point price elasticity of demand at the profit-maximizing level of output is     .

The level of fixed costs the firm is experiencing on its bed production is

.

What is the impact of a $5,000 increase in the level of fixed costs on the price charged, output produced, and profit generated?

	Increase	No change	Decrease
Price Charged
Output Produced
Profits Generated

Answer 1

Solution:

(Before going through the solution, please verify the equations once). We are given the (inverse) demand function as: P = 1,265 - 9Q

And the cost function is given as:

TC(Q) = Q^{​​​​​​3}/3 - 15Q² + 5Q + 24,000

Short-run profit maximizing condition is marginal revenue (MR) = marginal cost (MC)

Total revenue, TR = P*Q = (1265 - 9Q)Q = 1265Q - 9Q²

MR = $\partial TR/ \partial Q$ = 1265 - 2*9Q = 1265 - 18Q

MC = $\partial TC/ \partial Q$ = 3*Q^{​​​​​​2}/3 - 2*15Q + 5 + 0 = Q² - 30Q + 5

Then, using the profit maximizing condition:

MR = MC

1265 - 18Q = Q² - 30Q + 5

Q² - 12Q - 1260 = 0

On solving this quadratic equation, we get

Q = [-(-12) +/- ((-12)² - 4(1)(-1260))^1/2]/2(1)

Q = [12 +/- 72]/2

Q = 42 or Q = -30 (since, quantity cannot be negative, rejecting this negative value)

So, short run profits are maximized when level of output is 42 canopy beds.

And level of price, P = 1265 -9(42) = $887

At this output level, total cost = (42)^​​​​​³/3 - 15(42)² + 5(42) + 24,000

TC = 24696 - 26460 + 210 + 24000 = $22,446

TR = P*Q = 887*42 = $37,254

Profit = TR - TC

Profit = 37254 - 22446 = $14,808

Point price elasticity: e = $(\partial Q/ \partial P)*(P/Q)$

$\partial Q/ \partial P$ = (-1/9) (using the demand function:Q=(1265/9)-(1/9)Q)

So, at profit-maximizing point, with Q = 42 and P = 887,

e = (-1/9)*(887/42) = -2.35

In absolute terms, since elasticity is greater than 1, it is elastic demand curve at this (profit-maximizing) point.

Fixed cost is the cost incurred independent and irrespective of Q. From the total cost equation, we can see that the term independent of Q is the last one. Thus, here, level of fixed cost = $24,000.

If this fixed cost level increased by $5,000, then, TC increases by 5,000, but since there's no change of such increase in marginal functions, our profit- maximizing levels of output and price remains same.

Thus,

1) Price charged incurs no change

2) Output produced incurs no change

3) With total revenue remaining same, increase in total cost (due to increase in fixed cost), profits generated decrease.