Question

(Please answer this question in an excel format if possible.) Please provide very detailed explanation and answers if possible.

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 ¼ 1760 12Q

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

TC ¼ 1 3 Q3 15Q2 þ 5Q þ 24;000

a. Calculate the level of output that should be produced to maximize short-run profits.

b. What price should be charged?

c. Compute total profits at this price-output level.

d. Compute the point price elasticity of demand at the profit-maximizing level of output.

e. What level of fixed costs is the firm experiencing on its bed production?

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

(Please answer this question in an excel format if possible.)

Answer 1

SOLUTION:

PART (a) and (b)

The question was not very clear due to missing signs, but I assumed that 1/4 to actually mean '=', and proceeded further with solving the question. Also * sign indicates 'Multiplication', blanks means '-', and p stands for Plus.

Also, the answer is actually not meant to be solved in Excel, however, the typed answer should be helpful to you.

Given in the Question are the following informations relating to the demand and cost function of the Company.

P = 1760 - 12Q is the Demand curve of Price function

TC = [(Q^3)/3] - 15(Q^2) + 5Q + 24000, where TC = Total Cost

With the given demand curve, we can say that  Total Revenue (TR) = Price*Quantity

TR = (1760 - 12Q)*Q

TR = 1760Q - 12(Q^2)

Then, Marginal Revenue (MR) can be written as,

MR = (dTR)/(dQ)

MR = 1760 - 2*12Q

So, MR = 1760 - 24Q

Similarly, Marginal Cost (MC) = (dTC)/(dQ)

MC = (3/3)*[Q^(3-1)] - 2*15Q + 5

MC = [Q^2] - 30Q + 5

Profit maximizing level occurs at the quantity level where the MARGINAL COST and MARGINAL QUANTITY equate.

So, we must have MR = MC

1760 - 24Q = [Q^2] - 30Q + 5

[Q^2] - 6Q - 1755 = 0

Thus, solving the Quadratic Equation we get, Q = 45 or -39. Thus, we must consider Q = 45, as Q = -39 is not possible.

Since quantity can never be negative, the required output level is 45 beds.

Thus, The required PROFIT-MAXIMIZING Price is then: P = 1760 - 12*45 = $1220 per bed.

PART (c)

Profit = TR - TC

So, here Profit = [1760Q - 12(Q^2)] - [[(Q^3)/3] - 15(Q^2) + 5Q + 24000], where Q = 45

Profit = 54900 - 24225 = $30675

PART D

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

Now, since P = 1760 - 12Q

So, Q = (1760 - P)/12 or Q = 146.67 - 0.083*P

Then, dQ/dP = -0.083

And elasticity at this profit maximizing point (Q, P) = (45, 1220) is = -0.083*(1220/45) = -2.25.

PART E

The Fixed Cost is that part of TC curve which is independent of quantity (i.e. does not have a term that has a Q attached to it in any mathematical form).

Given the TC function, we can easily see that the only term independent of Q is 24000 (i.e. the last term in the TC curve).

So, TOTAL FIXED COST (TFC) on bed production is TFC = $24,000.

PART F

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

The profit maximizing quantity level is found at the intersection of marginal revenue and marginal cost curves. Also, further we can conclude that fixed cost (being independent of quantity) has no impact on marginal cost (as it is removed in the differentiation step from TC to MC) and thus any change in fixed cost will leave the MC function and curve unchanged. Of course MR 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, the PRICE CHARGED and OUTPUT PRODUCED has NO change, while the Profit generated decreases.

See below,

NEW Profit = TR - [TC + $5000] , where the $5000 is the increase in TFC.

So, we can say that

NEW Profit = [1760Q - 12(Q^2)] - [[(Q^3)/3] - 15(Q^2) + 5Q + 24000+ 5000] , where Q = 45

NEW Profit = 54900 - 29225 = $25675, i.e. New Profit reduced by $5000.