Applying the Perfectly Competitive Model

Applying the Perfectly Competitive Model.: the Spiffy Suit Manufacturers Problem

I. Application: Spiffy Suit Manufacturers, Inc. makes men's suits -- a (nearly) perfectly competitive activity. They are ready to begin production on their basic product for next season. The forecast price of these suits is $76 each (wholesale). Management believes it can rely on this figure. The estimated daily AVC function is

AVC = 100 -10Q +.5Q² where Q is suits per day

and AVC is in dollars

Total Fixed (per day) costs at Spiffy is $275

What should management do? (should they produce or not, if so what amount?, what will profits be?

Solution:

1) First find minimum AVC to see if price ($76) is above or below min. AVC.

min AVC is found by taking 1st deriv. of AVC function:

f' = -10 + 1Q

and setting it equal to 0: -10+Q = 0, Q = 10

at Q=10, AVC = 100 -10(10) + .5(10)² = 50 dollars,so: produce

2) What amount to produce? MC = P (or max Total Prof.)

a. the MC equation is the derivative of the total cost function, where:

TC = TVC + TFC , or TC = (Q X AVC) + TFC

thus TVC = 100Q - 10Q² + .5Q³

and TC = 275 + 100Q - 10Q² + .5Q³

the derivative (MC) = 100 -20Q + 1.5Q²

set that equal to 76 and solve for Q

76 = 100 -20Q + 1.5Q² , or 1.5Q² -20Q + 24 = 0

ans. via quad form.:Q= 1.33, 12 thus 12 suits

3) What profit? ans. TR-TC

a. TR is $76 X 12 = $912

b. TC is $275 + 100(12) -10(12)² + .5(12)³

which is 275 + 624 OR 899

note: (AVC at Q=12 is 100-120+.5(144) = 52 and 52X12=624)

c. so profits = $13 per day

4) Alt. method: maximize Tot. Prof. function:

TR - TC = Total Profit

PxQ - (275 + 100Q -10Q² + .5Q³)

76Q -275 -100Q +10Q² - .5Q³ ,or

Total Profit = -275 -24Q + 10Q² -.5Q³

find 1st deriv. f'= 24+20Q-1.5Q² set equal to 0

solve for Q (using quadratic form. as above) Q =12

Then substitute Q=12 in total profits function, getting

-275 - 24(12) + 10(12)² - .5(12)3

Total profit = 14

II.. Implementing Monopoly Theory -- Finding the Profit Max. Price and Quant.

A demand function of the usual sort will generally apply. For example:

Q = a + bP + cI +dP_R

Management will have to obtain projected estimates of I (income) and P_R (price of other good).

Once that is done the I and P_R terms collapse into the constant (a').

Say,in a practical case that the eqn is Q = 99 -2P + .01I + .1P

and the estimates for I and P_R are 10,000 and 10 respectively

Our equation becomes Q= 200 -2 P

Next, we can derive a MR equation from the demand eqn., but first put it in inverse form

P = 200/2 - 1/2 Q

MR will resemble the (straight line) demand curve but will decend twice as fast.

thus MR = 100 - Q

Next MR will be equated with MC, where, again, MC can be derived from the AVC curve (it is the derivative of the TVC curve).

If the AVC curve is AVC = 5 -5Q + Q²