Chapter 8: Profit Maximization by a Competitive Firm
1. True , false , or uncertain. Explain.
a. A firm necessarily maximizes profit or minimizes losses when the difference between P and AC is
maximized.
b. Price and marginal revenue are the same for a competitive firm.
c. A firm that produces a quantity at which MR = MC is maximizing TR.
d. A firm that continues to produce when it is incurring losses is not behaving rationally.
e. A firm’s short-run supply curve is that section of the short-run marginal cost curve that lies above its
average cost curve.
f. The long-run industry supply curve is more elastic than the short-run industry supply curve for
constant-cost industries , but not for increasing-cost industries.
g. An increase in market demand will lead to a larger price increase in the short run than in the long
run.
h. An increase in market demand will lead to a larger market output increase in the short run than in the
long run.
i. A 10% profit tax will have the same impact on firm output as an excise tax.
j. A increase in variable costs will lead to a larger price increase in the long run in constant-cost industries
than in increasing-cost industries
2. A firm faces the following total cost function: TC = $3000 + 2Q + 0.1Q 2 .
Output sells competitively at $42/unit.
a. Express TR and profit as a function of Q.
b. What is the MC function? What is AC function? What is the firm’s short run supply curve?
c. What if the profit-maximizing rate of output? What are total profits at this output rate?
d. Suppose fixed costs rise from $3000 to $3500. What happens to the TC function ?
...to the MC and AC functions? What happens to profit-maximizing output? ...to total profit?
e. If fixed costs instead increased from $3000 to $6000, would a firm continue to operate in
the short run? (hint: find VC)... in the long run?
f. A tax of $10 per unit is imposed on this firm. Find the new TC, MC and AC functions.
What happens to the profit-maximizing rate of output?
3. Analyze the effects on a typical firm and the industry of the following changes. Give both
the short run and long run effects. Provide both graphs and a brief written explanation.
a. Firms face lower insurance premiums (fixed costs). Assume an increasing-cost industry.
b. Firms face an increase in raw material prices (variable cost). Assume a constant-cost
industry.
c. Improved technology enables firms to produce any given quantity of output using fewer
variable inputs. Assume a constant-cost industry.
4. Use the following variable cost function to answer the questions below:
VC = 50Q - 12Q2 + 2Q3 . Fixed Costs = $10.
a. What is the TC function? What will a graph of the VC and TC curves look like?
b. Find the MC , AVC, and AC equations.
c. Suppose FC increase to $20. What happens to the VC equation? ...to the TC equation?
...to the MC equation? ...to AVC equation? ...to the AC equation? How would you show this increase on a
graph?
d. Suppose the a tax of $10 per unit is imposed on this firm. Answer the same questions in part (c).
ANSWERS
1. a. False. This only means that profit per unit is maximized, which occurs at the quantity where AC is
minimized. Total profit is maximized at the quantity where MR = P = MC which may or may not be at
the quantity where AC is minimized.
b. True. A competitive firm (price-taking firm) can sell all they want at the going market price. This
implies that P equals MR.
c. False. This is the condition for maximizing total profit. As long MR>0, a firm can increase its TR by
producing another unit of output. This being the case, TR is therefore not at its maximum.
d. Uncertain. If P is less than AC (and hence incurs losses), but greater than AVC then it is rational to
continue to operate in the short run. If it shut down, the firm’s losses would be even larger. On the other
hand, if P is below AVC and the firm continues to operate in the short run, then it is behaving irrationally.
e. False. It is the section of the MC curve that lies above AVC.
f. False. The long run industry supply curve is more elastic than the SR industry supply curve in either type
of industry.
g. True. Since the SR supply curve is more inelastic than the LR supply curve (for any type of industry),
then it follows that a demand increase will lead to a larger price increase in the SR than in the LR.
h. False. See explanation given in part (g).
i. False. At profit tax will not change the profit-maximizing output, but an excise tax (which affects MC)
will.
j. True. Since the LR supply curve is perfectly elastic in a constant-cost industry, and since the LR
industry supply curve is less than perfectly elastic in an increasing-cost industry, it follows that for a given
increase in variable costs, price will increase by more in the constant-cost industry than in an increasingcost industry. See graph.
2a. TR = 42Q and total profit = 42Q - 3000 - 2Q - 0.1Q2 = 40Q - 3000 - 0.1Q2.
b. The MC function is found by taking the derivative of the TC function with respect to Q. We get:
MC = 2 + 0.2Q (notice that this gives us a straight line for the MC curve).
To find the AC function, divide both sides of the TC function by Q. You should get:
AC = 3000/Q + 2 + 0.1Q.
The supply curve is an equation which relates Q to P. Since profit-maximization occurs where P = MC, it
follows that we can substitution P for MC in the above MC equation. This gives us the following supply
curve: P = 2 + 0.2Q (this is actually the inverse supply function; if we solve for Q we get the supply
function -- Q = -10 + 5P).
c. The profit-maximization condition is MR = MC. Since P equals MR for a price taker, MR = 42.
Set this equal to MC and solve for Q. 42 = 2 + 0.2Q and solving for Q gives us Q= 200 units.
Total profit = (P-AC) (Q) . AC = 3000/200 + 2 + 0.1(200) = 37, therefore total profit = (42-37)(200) =
1000.
d. The new TC function is 3500 +2Q + 0.1Q2. The MC is unchanged and the new AC is
3500/Q +2 + 0.1Q. Since MR and MC curves are unchanged there is no change in profit-maximizing
output. To find total profit we need to find AC at 200 units. AC = 3500/200 + 2 + 0.1(200) = 39.5
Total profit = (42-39.5)(200) = 500.
e. First, notice that a firm would still produce at 200 units if it was to produce at all. This follows from the
fact that a change in fixed costs does not affect the MC curve. Next, notice that this firm is no longer able
to cover its total costs when FC = 6000 (TR = 8400< TC = 10,400; or alternatively, P =42< AC=52).
However, the firm can cover its variable cost (TR =8400> VC = 4400; or alternatively, P= 42> AVC = 22).
Therefore, the firm will continue to operate in the short run.
The firm will stop operating in the long run. This is because all costs are variable costs in the long run and
this firm is unable to cover its all of its costs.
f. The tax of $10 per unit will, for any output Q, add 10Q cost, so the TC cost function becomes
TC = 3000 + 2Q + 0.1Q2 + 10Q = 3000 + 12Q + 0.1Q2.
MC = 12 + 0.2Q (notice that this shifts the MC curve up by a vertical amount equal to $10.)
AC = 3000/Q + 12 + 0.1Q.
To find the new profit-max. rate of output:
MR = MC
42 = 12 + 0.2Q ---> Q = 150.
3. See graphs.
$ per unit
Question #1a.
MC
AC
P
MR=D
Q
Q1 Q2
P-AC is maximized at Q1, but TOTAL profit is maximized at Q2.
P
(P-AC) = profit per unit.
Question 1j.
LRS2increasing-cost
LRS1increasing-cost
P2c-c
P2i-c
LRS2constant-cost
P1
LRS1constant-cost
D1
Q2 Q2 Q 1
Q
The price increase in the LR is larger in the constant-cost case than in the increasing-cost case.
P
Question 1g-example for increasing-cost industry
SRS1
PSR
LRS1
PLR
P1
sr
lr
D1
QSR QLR
Q1
D2
Q
QUESTION 3A.
Firm
Market
$ per unit
n
MC=SRSF
SRSM=SRSF
AC1
LRS1
AC3
P1=MC1=AC1
P1
AC2
LRS2
PLR
PLR
DM
q
qLR
Q
q1 =qSR
Q1=QSR
QLR
QUESTION 3B.
Firm
AC2
Market
SRSM2=SRSF2
SRSM1=SRSF1
$ per unit
MC2
ACSR
PLR
MC1=SRSF1
AC1
PLR
LRS2
PSR
MRSR
P1=MC1=AC1
PSR
MR1=D1 P1
DM
q
qSR
q1=qLR
Q
QLR
QSR
Q1
4. a. TC = VC + FC = 10 + 50Q - 12Q2 + 2Q3.
b. MC = dTC/dQ = dVC/dQ = 50 - 24Q + 6Q2.
AVC = VC/Q = 50 -12Q +2Q2 and AC = TC/Q = 10/Q + 50 - 12Q + 2Q2 .
c. No change in VC function. TC = 20 + 50Q -12Q2 + 2Q3.
No change in MC function or the AVC function.
AC = 20/Q + 50 - 12Q + 2Q2.
The FC and TC curves would shift by a vertical amount equal to the increase in fixed costs.
No change in the VC, MC, or AVC curves. AC shifts up.
d. At any given output, costs are now greater by 10Q. Hence:
TC = 10 + 50Q + 10Q - 12Q2 + 2Q3 = 10 + 60Q - 12Q2 + 2Q3.
VC = 60Q - 12Q2 + 2Q3.
MC = 60 - 24Q + 6Q2 .
AVC = 60 -12Q + 2Q2 and AC = 10/Q + 60 - 12Q + 2Q2.
Notice that the MC, AVC and AC curves all shift by a vertical amount equal to 10 at any level of Q.
Question 4a.
VC, TC
TC
VC
FC=$10
FC
Q
True or False questions.
1. In the short run, if the price of capital (fixed input) increases, the firm’s short run AC, AVC , and MC
curves will all be unaffected.
2. In the short run, if the price of labor (variable input) increases, the firm’s short run AC, AVC, and
MC curves will all be unaffected.
3. If the short run MC curve is increasing, then the AVC curve must also be increasing.
4. The vertical distance between the AVC curve and the AC curve is the same at all levels of
output. Assume that FC > 0.
5. At a given level of output, if short run total costs are minimized, then long run total costs
must also be minimized.
6. Suppose a firm experiences constant returns to scale at all levels of output. It therefore follows that when
the firm increases its use of all inputs, its output expands proportionately. Thus this firm
never experiences diminishing marginal returns to labor.
True or False questions-answers.
1. False. AC ,which equals AFC + AVC, will shift upwards. It is true that a change in fixed costs will not
affect the AVC or MC curves.
2. False. All of the curves will shift up.
3. False. MC may possibly be rising but still fall below the AVC curve. If MC is below AVC, the AVC
must be falling.
4. False. Since AC = AVC + AFC, then the gap between AC and AVC equals AFC:
AC - AVC = AFC. But AFC is always getting smaller as Q increases, and therefore the gap between AC
and AVC must also be falling.
5. False. Since at least some inputs are fixed in the short run, then the optimal combination of L and K in
the long run (when all inputs are variable) may be different from the combination of L and K used in the
short run.
6. False. The principle of diminishing marginal returns to labor is a short run phenomenon since it says that
the marginal product of a variable input will eventually diminish holding other inputs fixed. Returns to
scale refers to the impact on output when all inputs are varied.