Question

Homework (Ch 10) 5. 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 = 3,005 - 100 The cost analysis department has estimated the total cost function for the poster bed as TC = - 1502 +50 +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 s 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 s -4.01 What is the impact of a $5,000 increase in the level of fixed costs on the price -0.02 output produced, and profit generated? -0.10 Increase No change Decrease

Answer 1

Solution:

We are given the inverse demand function as:

P = 3005 - 10Q

Total cost function is given as: TC = Q3/3 - 15Q2 + 5Q + 24000

With given price function, we know total revenue = price*quantity

TR = (3005 - 10Q)*Q

TR = 3005Q - 10Q2

Then, marginal revenue = differentiation of total revenue with respect to quantity

MR = dTR/dQ

MR = 3005 - 2*10Q

So, MR = 3005 - 20Q

Similarly, marginal cost = dTC/dQ

MC = 3*Q2/3 - 2*15Q + 5

MC = Q2 - 30Q + 5

Profit maximizing level occurs at quantity level where the marginal cost equals the marginal revenue. So, we must have MR = MC

3005 - 20Q = Q2 - 30Q + 5

Q2 - 10Q - 3000 = 0

(Q - 60)(Q + 50) = 0, thus Q = 60 or -50

Since, quantity can never be negative, the required output level is 60 beds. The required profit maximizing price is then: P(10) = 3005 - 10*60 = $2405 per bed.

Profit = total revenue - total cost

So, here profit = 2405*60 - (603/3 - 15*602 + 5*60 + 24000

Profit = 144300 - 42300 = $102,000

Point elasticity of demand = (dQ/dP)*(P/Q) at point (Q, P)

Now, P = 3005 - 10Q

So, Q = (3005 - P)/10 or Q = 300.5 - 0.1*P

Then, dQ/dP = -0.1

And elasticity at this profit maximizing point (Q, P) = (60, 2405) is = -0.1*(2405/60) = -4.008 or -4.01 (approximately).

The fixed cost is that part of total cost which is independent of quantity. Given the total cost function, we can easily see that only term independent of Q is 24000. So, total fixed cost faced by the firm on bed production is $24,000.

With an increase in level of fixed cost by $5000, new total fixed cost = 24000 + 5000 = $29,000

Notice that the profit maximizing quantity level is found at intersection of marginal revenue and marginal cost curves. Further note that fixed cost (being independent of quantity) has no impact on marginal cost and thus any change in fixed cost will leave marginal cost function and curve unchanged. Of course marginal revenue is unchanged (change in cost will not affect revenue related functions in any way).

Thus, the profit maximizing output level produced and price level charged will stay unchanged. However, profits are affected by fixed costs, one-to-one negatively. So, increase of $5000 in fixed cost will reduce profits by $5,000.

So, price changed and output produced face no change, while profit generated decreases.