ECON 2001
Microeconomics II
2017 2nd semester
Elliott Fan
Economics, NTU
Lecture 7
Microeconomics, 2017 Spring
Elliott Fan
Lecture 6
Profit maximization
max pf ( x1 , x2 )  w1 x1  w2 x2
x1 , x2
The FOCs are:
f ( x1* , x2* )
f ( x1* , x2* )
p
 w1 , p
 w2
x1
x2
Assuming Cobb-Douglas production function
given by f ( x1 , x2 )  x1a x2b , then the FOCs reduce to:
pax1a 1 x2b  w1  0
pbx1a x2b 1  w2  0
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Profit maximization
Multiple the first equation by x1 and the second
equation by x2 , then the FOCs reduce to:
pax x  x1w1  0 and pbx x  x2 w2  0
a b
1 2
a b
1 2
which are:
pay  x1w1  0
and
pby  x2 w2  0
So we now have:
apy
bpy
*
*
x1 
, x2 
w1
w2
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Profit maximization
The remaining question is to derive supply funciton of y.
To do so, we insert the two factor demand functions into
the production function, then we have:
apy a bpy b
yx x (
) (
)
w1
w2
a b
1 2
so,
ap
y( )
w1
a
1 a b
bp
( )
w2
b
1 a b
We can then substitute this into the factor
demand functions.
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Section 15.1
THE FIRM’S DEMAND FOR FACTORS
IN THE LONG RUN
5
Recap Production and Costs in the
Long Run
• Firm can adjust employment of capital and labor
– Achieve the least cost method of producing a
given quantity of output
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Isoquants
• Geometry of LR production
– Requires labeling vertical axis with K, stands for capital
– Requires labeling horizontal axis with L, which stands for labor
– Requires fixed period of time
• Least costly method
– Avoid technologically inefficient points which are outside the
boundary
• General observations about isoquants
–
–
–
–
Slope downward
Fill the labor-capital plane
Never cross
Convex to origin
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Marginal Rate of Technical
Substitution
• Absolute value of slope of isoquant
– MPL divided by MPK
• Amount of capital necessary to replace one unit
of labor while maintaining a constant level of
output
– If much labor and little capital employed to
produce a unit of output, MRTSLK is small
• Provides geometric proof that isoquant is convex
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Marginal Rate of Technical
Substitution
• The discussion above assumed a one-unit change
in labor. More generally, if labor changed by
some amount of ∆L, we will have:
L  MPL  K  MPK
and we would have:
MRTS LK
K MPL


L MPK
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Choosing a Production Process
• Minimizing cost necessary for maximizing profit
• Isocost curve
– Tracks set of all baskets of inputs employed
– Assume cost fixed
– Slope: -PL/PK
• Firm chooses point where isocost and isoquant
curves tangent
– Means MRTS = PL/PK
• Tangencies lie along firm’s expansion path
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13
Firm’s Demand in the LR
• All factors variable
• Assume fixed technology (the production
function), rental rate (PK), and market price (PX).
• Note that making the assumption that PK is fixed
incurs no loss of generosity, as only the relative
price matters.
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Construction of LR Labor Demand
Factor demand vs output demand:
• The major difference is that a firm, unlike the
case of output demand, has no budget constraint.
Instead it has an infinite family of isocost lines,
and it could choose to operate on any one of
them.
• So we call factor demand “derive” from output
demand. In short, we have to consider the
optimal decision on the output market.
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Construction of LR Labor Demand
• To be more precise, we need to determine how
much to produce before we determine exactly
how much factors to hire. Eg:
apy
bpy
*
x 
, x2 
w1
w2
*
1
• We, again, need to resort to the principle of
MR=MC.
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18
Construction LR Labor Demand
Substitution and scale effects associated with a
factor price change
• SubE: When the price of an input changes, that
part of the effect on employment that results
from the firm’s substitution toward other inputs.
• ScaE: When the price of an input changes, that
part of the effect on employment that results
from changes in the firm’s output
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Substitution and Scale Effects
Direction of substitution effect and scale effect
• Always reduces firm’s employment of labor
• An increase in the wage rate raises firm’s long-run
total cost curve:
– Could rise and become steeper, causing longrun marginal cost to rise.
– Could rise and become shallower, causing
long-run marginal cost to fall when labor is a
regressive factor.
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21
22
Substitution and Scale Effects
• Combine effects
– Labor demand curve always slopes downward
– The proof is available at the appendix
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SR and LR Relationship
• In LR
– MRP shifts due to adjustments in capital
employment
• Infinite number of steps
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Section 15.3
THE INDUSTRY’S DEMAND FOR
FACTORS OF PRODUCTION
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Industry’s Demand
• Sum of individual firm’s demand curve for factor
of production
• Monopsony
– Upward-sloping supply curve
– Marginal labor cost (MLC)
– Employment and wage rate
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Industry’s Demand
• Existence of monopsony
– Even a firm that is unique in its industry has no
monopsony power, provided that firms in
other industries compete with it for the use of
the factors.
– Monopsony is rare, especially in the long run.
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Section 15.4
THE DISTRIBUTION OF INCOME
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Distribution of Income
Payments to factors of production
• In Figure 15.11, total revenue is A+B+C, of which
B+C are paid to workers.
• C refers to the opportunity costs for workers.
• B is earned as rent.
• A is payment to other factors, including the
owner’s contribution.
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33
Distribution of Income
Returns to scale
• Decreasing return to scale (DRS) – AC curve is
increasing – Price is higher than AC (why?) –
positive profit.
• IRS – negative profit.
• CRS – zero profit.
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Distribution of Income
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Distribution of Income
Returns to scale
• Decreasing return to scale (DRS) – AC curve is
increasing – Price is higher than AC (why?) –
positive profit.
• IRS – negative profit.
• CRS – zero profit.
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Distribution of Income
Who benefits?
• Recall the tax incidence.
• Compare the cases of tax and subsidy.
• Now we talk about rent in the case of factor
supply.
• Bottomline: factors that are supplied relatively
inelastically earn more rents and gain more from
a rise in demand.
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