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ECON 3039
Labor Economics I
2013
By Elliott Fan
Economics, NTU
Labor demand
Economics of Labor, 2013
Elliott Fan
Lecture 4
Firm’s Demand in SR
• Only labor input is variable while capital amount is fixed.
• Marginal revenue product (MRP)
– Additional revenue earned if employ one more unit of input
– When factor price is fixed, MRP is obtained by marginal product
of labor times marginal revenue of the output. (you need to be
careful on this)
MRPL  MRX  MPL
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Firm’s Demand in SR
Wage rate and MRP
• Marginal product of labor is defined as the contribution to
output by adding an additional unit of labor.
• So the amount of labor needed to produce one more unit of
output is:
1 / MPL
• And the cost of this amount is (this, by definition, refers to
marginal cost of x):
PL / MPL
Economics of Labor, 2013
Elliott Fan
Lecture 4
Firm’s Demand in SR
Wage rate and MRP
• Now we know that when the firm maximizes profit, they set
marginal revenue equal to the marginal cost:
• So:
MRX  MC X  PL / MPL
PL  MRX  MPL  MRPL
Economics of Labor, 2013
Elliott Fan
Lecture 4
Firm’s Demand in SR
Two major points drawn from the algebra:
• It confirms that a profit-maximizing firm operates where
the wage rate of labor is equal to its marginal product of
labor.
• The firm’s demand curve for labor coincides with the
MRPL curve.
Economics of Labor, 2013
Elliott Fan
Lecture 4
Some comparative statics
• Wage goes up.
• Output price goes up.
• An increase in plant size
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Some comparative statics
• Wage goes up.
• Output price goes up.
• An increase in plant size
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Some comparative statics
• Wage goes up.
• Output price goes up.
• An increase in plant size
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
which are:
pay  x1w1  0
a b
1 2
and pby  x2 w2  0
So we now have:
apy
bpy
*
*
x1 
, x2 
w1
w2
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
a b
y  x1 x2  (
) (
)
w1
w2
so,
ap
y( )
w1
a
1 a b
bp
( )
w2
b
1 a b
We can then substitute this into the factor
demand functions.
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
K MPL


L MPK
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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.
Economics of Labor, 2013
Elliott Fan
Lecture 4
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 “derived” from output
demand. In short, we have to consider the optimal
decision on the output market.
Economics of Labor, 2013
Elliott Fan
Lecture 4
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.
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
Substitution and Scale Effects
• Direction of substitution effect
– Reduces firm’s employment of labor
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
Substitution and Scale Effects
• Direction of scale effect
– LRTC rise and shallower
 LRMC rises
– Regressive factor
• Combine effects
– Labor demand curve always slopes downward
– Scale effect never dominates substitution effect
– The proof is here.
Economics of Labor, 2013
Elliott Fan
Lecture 4
SR and LR Relationship
• In LR
– MRP shifts due to adjustments in capital employment
• Infinite number of steps
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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
Economics of Labor, 2013
Elliott Fan
Lecture 4
Economics of Labor, 2013
Elliott Fan
Lecture 4
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.
Economics of Labor, 2013
Elliott Fan
Lecture 4