Profit Maximization

Profit Maximization
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The Nature and Behavior of Firms
• Simple model of a firm
– Technology given by the production function,
f(k, l)
• Inputs: labor (l) and capital (k)
– Run by an entrepreneur
• Makes all the decisions
• Receives all the profits and losses from the firm’s
operations
• Acts in his or her own self-interest
– Maximize the firm’s profits
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Profit Maximization
• A profit-maximizing firm
– Chooses both its inputs and its outputs
• With the sole goal of achieving maximum economic
profits
– Seeks to maximize the difference between
total revenue and total economic costs
– Make decisions in a “marginal” way
• Examine the marginal profit obtainable from
producing one more unit of hiring one additional
laborer
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Profits
Definition: Economic Profit
Sales Revenue - Economic (Opportunity) Cost
Example:
• Revenues: $1M
• Costs of supplies and labor: $850,000
• Owner’s best outside offer: $200,000
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The Profit Maximization Hypothesis
“Accounting Profit”: $1M - $850,000 = $150,000
“Economic Profit”: $1M - $850,000 - $200,000 = -$50,000
• Business “destroys” $50,000 of owner’s wealth
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Shut Down Rule
The firm will choose to produce a positive output q only if:
(q) > (0) …or…
Pq – TVC(q) – TFC > -TFC 
Pq – TVC(q) > 0 
P > AVC(q)
Definition: The price below which the firm would opt to
produce zero is called the shut down price.
Because there are FC, we know this is the short run. If
P<AVC the firm produces nothing, hence shuts down
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Marginal Revenue
Marginal revenue =dTR/dQ=dpq/dq
dpq dp
dq dp

q p

q p 
dq dq
dq dq

1
 MR  p 1 
 e
p

 dp q 
p
 1
 dq p 

  p since e p  0

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Example, MR from a Linear Demand Function
Demand curve for a sub sandwich is q = 100 – 10p
Solving for price: p = -q/10 + 10
Total revenue: R = pq = -q2/10 + 10q
Marginal revenue: MR = dR/dq = -q/5 + 10
Compare MR and price: 10-q/5<10-q/10 so
MR < p for all values of q
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Marginal Revenue and Elasticity I
• Marginal revenue is closely related to the
elasticity of the demand curve facing the
firm
• The price elasticity of demand = %∆q/%∆p
dq / q dq p
ep 
 
dp / p dp q
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Marginal Revenue and Elasticity II
• Since we have
 q dp 
q  dp
MR  p 
 p 1    
dq
 p dq 
 1
p 1  
 e 
p 

– Demand slopes downward so ep < 0 and MR < p
– If demand is elastic: ep < -1 and MR > 0
– If demand is unit elastic: ep = -1 and MR = 0
– If demand is inelastic: 0>ep > -1 and MR < 0
– If demand is infinitely elastic: ep = - and MR = p
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11.2
Market Demand Curve and Associated Marginal Revenue
Curve
Price
p2
p1
D
q2
q1
Quantity per period
MR
If the demand curve is negatively sloped, so the marginal revenue curve will fall below the
demand (‘‘average revenue’’) curve. For output levels beyond q1, MR is negative. At q1, total
revenues (p1 · q1) are a maximum; beyond this point, additional increases in q cause total
revenues to decrease because of the concomitant decreases in price.
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Output Choice
• Total revenue for a firm, R(q) = p(q)q
• Economic costs incurred, C(q)
• Economic profits, 
– The difference between total revenue
and total costs
(q) = R(q) – C(q) = p(q)q –C(q)
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Output Choice
The necessary condition for choosing the level of q
that maximizes profits requires setting the derivative
of the  function with respect to q equal to zero, so
choose q so that
d
dTR dTC
  '(q) 

0
dq
dq
dq
dTR dTC


 MR  MC
dq
dq
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Second-Order Conditions (not for testing)
MR = MC is only a necessary condition for
profit maximization
It is also required:
d 2
dq 2
d '(q)

0
dq q  q*
q  q*
‘‘marginal’’ profit must decrease at the optimal level
of output, q*
– For q<q*, ′(q) > 0
– For q>q*, ′(q) < 0
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11.1 (a)
MR=MC for Profit Maximization: Graphical interpretation
Revenues,
Costs
C
R
q**
q*
Output per period
•Profits reach a maximum when the slope of the revenue function (marginal revenue) is equal to
the slope of the cost function (marginal cost). This equality is only a necessary condition for a
maximum, as may be seen by comparing points q* (a true maximum) and q** (a local minimum),
points at which marginal revenue equals marginal cost.
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11.1 (b)
MR=MC for Profit Maximization
Profits
q*
Output per period
Losses
•Profits, defined as revenues (R) minus costs (C), reach a maximum when the slope of the revenue
function (marginal revenue) is equal to the slope of the cost function (marginal cost). This equality
is only a necessary condition for a maximum, as may be seen by comparing points q* (a true
maximum) and q** (a local minimum), points at which marginal revenue equals marginal cost.
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Revenues,
Costs
C
q**
q*
q**
q*
R
Output per period
Profits
Output per period
Losses
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The Profit Maximization Condition
MC=MR
MC is rising
(This MR curve assumes ep=-)
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Cost-plus pricing
• Many businesses price by a “mark-up” over cost.
• They figure out the cost per unit, and add a
percentage to that price
• Profit maximization is consistent with “cost-plus”
pricing
• It consists of setting price above MC
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Marginal Revenue
Already showed
dTR d [ p (q )  q ]
dp
MR 

 p  q
dq
dq
dq
and so
 q dp 
MR  p 1    
p dq 


1
p 1 
 e
p




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Price–Marginal Cost Markup
Maximize profits: MR = MC so

1
MC  p 1   , or
 e 
p 

p  MC
1
1


p
e p e p
since demand is downward sloping and thus ep < 0
• Recall we argued that a firm will produce where demand is
elastic, so ep<-1 (or equivalently, |ep|>1)
• The percentage markup over marginal cost will be higher
the closer |ep| is to 1
• Given that |ep|>1, what is the highest percentage markup
over MC for a profit maximizing firm?
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Price-Average Cost Markup
• If there are constant economies of scale
AC=MC=constant
– To see, AC=TC/q
MC=dTC/dq
– If there are constant economies of scale, AC=constant.
– But we know if MC>AC, AC is increasing, and if
MC<AC, AC is decreasing
– Hence, for AC to be constant, AC=MC
• We just showed profit maximization implies
p  MC
1
1


p
e p e p
• Hence in constant economies of scale MC=AC and
p  MC p  AC
1


p
p
ep
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Cost Changes and Profit Maximization
• This is the short run, so generally we would only think about changes
in variable cost
• But it is actually possible for fixed costs to change in the short run –
for example, a firm can be hit with a higher property tax all of a
sudden
• Changes in FC have no effect on short term decisions
max   TR(q)  TC (q)  TR(q)  VC (q)  FC
d dTR dTC dTR dVC




0
dq
dq
dq
dq
dq
 MR  MC
• Since FC don’t change with output, a change in FC has no effect on
optimal output choice in the short run; it depends only on revenues
and variable cost
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Cost Changes and Profit Maximization
• Changes in costs will have an effect on short term decisions
• If the cost curve shifts so MC decreases at an give output, will q goes up
– At any q, TC curve C2 has a flatter slope than C1, so MC2<MC1
– Profit maximizing output goes up to q**
• Why are we looking only at short run?
Revenues,
Costs
C1
R
C2
FC
q*
q**
Output per period
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Revenue Changes and Profit Maximization
• Changes in TR will also have an effect on short term decisions
• If the TR curve shifts so MR increases, so will output
– TR curve R2 has a steeper slope than R1, so MR2>MR1
– Profit maximizing output goes up to q**
R2
Revenues,
Costs
C
R1
FC
q* q**
Output per period
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Revenue Maximizing firms
• Because of the nature of executive
compensation and because revenue is
easier to measure than profit, some
analysts believe firms often maximize
revenue subject to some minimum level of
acceptable profit
• For example, market share is usually
measured in terms of revenue, and
executive compensation is often tied to
market share, so CEOs pay attention to
revenue
• Generally, a revenue maximizing firm will
produce more than a profit maximizing
firm
• A revenue maximizing firm with a
constraint that 0 will produce at Q2>Q
where the profit maximizing firm would
produce
Q
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Math for the revenue maximizing firm
• A revenue maximizing firm with a profit constraint will produce where
MR=0 unless that output puts profit below the acceptable level
max TR 
dTR
 0  MR  0
dq
• More generally we set up a lagrangian
max L  TR   (   *)  max L  TR   (TR  TC   *)
where * is the minimum acceptable profit. Doing the math we find
dL

 0  MR   MR   MC  0 so MR 
MC
dq
1 
It can be shown   0 and we know MR  0 (for TR to be maximized)
Hence  1 

1 
 0. So MR  MC
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An example
Profit maximizing
Let p  100  q and TC  500  q  q 2 
  q(100  q)  500  q  q 2 
1 3
q
30
1 3
q
30
1 2
max   100  2q  1  2q  q  0
10
q 2  990  0 so q  990  31.46
  1577
Revenue Maximizing
MR  0  100  2q  0 so q  50
  283
If it is required that   1000 we need
1
(100  q)  500  q  q 2  q 3  1000 
30
qR  47 (at q  47   1192; q  48   565)
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