Question

Suppose that a price-setting firm has the following total revenue and total cost functions:

R(q) = 10.75q – 0.1875q2 and C(q) = 75 + 0.07q + 0.035q2 .

This firm faces downward sloping demand and marginal revenue curves. Marginal revenue and marginal cost are given by

?? ??
= ??(?) = 10.75 – 0.375? and    ?? ??
= ??(?) = 0.07 + 0.07?,

respectively.

a. Using the marginal revenue function given above, find an expression for the firm’s demand curve
as a function of q. I.e., D(q) = p = ??? b. What is the firm’s profit maximizing output level? I.e., ??? ??? ?: ?(?) = ?(?) −?(?).
c. Since the firm is a price setter, rather than a price taker, what price will it need to set in order to
achieve the profit maximizing output level from b. above?
d. Using the answers you found in b. and c. above, what is the firm’s total revenue? e. Using the results you found above, find the firm’s inverse demand function. I.e., D-1(p) = q = ???
f. Using the results you found from b., c., and e. above, what is the price elasticity of demand (i.e., ε
= ? ?
∙ ?? ?? ) at the firm’s profit maximizing output level? Is the profit maximizing output level in the
elastic range or is it the inelastic range of the firm’s demand curve? In which of the two ranges
should we expect a price setter to operate?
g. Use the open source “Graph” software (version 4.4.2 downloadable at http://www.padowan.dk/)
to plot the total revenue function, the cost function and the profit (π = TR – TC) function. Print
copies of your graphs and attach them to this quiz or email them to [email protected]. If you
prefer some other graphing software, such as https://www.desmos.com/calculator, use it rather
than Graph.

Answer 1

Solution:

We are given the

total revenue function, R(q) = 10.75q - 0.1875q²

And, total cost fubction, C(q) = 75 + 0.07q + 0.035q²

The corresponding marginal revenue, MR(q) = 10.75 - 0.375q

And marginal cost function, MC(q) = 0.07 + 0.07q

a). Note that total revenue, R(q) = price*quantity = p*q

So, p = R(q)/q

p = (10.75q - 0.1875q²)/q = 10.75 - 0.1875q

The inverse demand function thus is: p(q) = 10.75 - 0.1875q

b) Profit-maximizing output level for a price setter firm exists where MR(q) = MC(q)

Then, 10.75 - 0.375q = 0.07 + 0.07q

q = (10.75 - 0.07)/(0.07 + 0.375)

q = 10.68/0.445 = 24 units

c). We need to find the price level charged at the above found profit maximizing output level. Using the inverse demand curve, we find

p = 10.75 - 0.1875*24 = 10.75 - 4.5

p = $6.25

d). Firm's total revenue = p*q = 6.25*24 = $150

e). Finding the demand curve:

We have inverse demand curve as: p = 10.75 - 0.1875*q

Then, 0.1875q = 10.75 - p

On dividing both sides by 0.1875, we get

q = (10.75 - p)/0.1875

q = 57.33 - 5.33*p. This is the required demand function

f). Price elasticity of demand, ed = *(p/q)

= -5.33

So, ed = (-5.33)*(6.25/24) = -1.389

Since, |ed| = |-1.389| = 1.389 > 1, profit maximizing output is indeed in the elastic range of demand curve.

Price setter is expected to operate in this elastic range only, since having the market power, if it operates on the inelastic range, it will always have an incentive to increase price further which will further increase the total revenue (since, inelastic, so price and total revenue move in same direction) and this way, a maximum will never be reached.