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Short-run Production Function
• Describes the technology that the firm uses to produce goods and services
– E.g.,
q  K E
• The more E and K the higher the firm’s output
K
E
q
0
0
0
100
100
100
200
200
200
300
300
300
400
400
400
500
500
500
Short-run Production Function
• Over the long-run K varies, but in the short-run K is fixed
– E.g., K = 400 and
q  400  E
q  20 E
• The more E the higher the firm’s short-run output
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
Law of diminishing marginal productivity
•
The marginal product of labor is (MPL) the change in output resulting
from hiring an additional worker, holding constant the quantities of other
inputs
MPL 
E  100  0  100
q 200

2
E 100
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
q  200  0  200
Law of diminishing marginal productivity
•
The marginal product of labor is (MPL) the change in output resulting
from hiring an additional worker, holding constant the quantities of other
inputs
MPL 
E  200 100  100
q 83

 0.83
E 100
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
q  283  200  83
Law of diminishing marginal productivity
•
The marginal product of labor is (MPL) the change in output resulting
from hiring an additional worker, holding constant the quantities of other
inputs
MPL 
E  300  200  100
q 63

 0.63
E 100
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
q  346  283  63
Law of diminishing marginal productivity
•
The marginal product of labor is (MPL) the change in output resulting
from hiring an additional worker, holding constant the quantities of other
inputs
MPL 
E  400  300  100
q 54

 0.54
E 100
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
q  400  346  54
Law of diminishing marginal productivity
•
The marginal product of labor is (MPL) the change in output resulting
from hiring an additional worker, holding constant the quantities of other
inputs
MPL 
E  500  400  100
q 47

 0.47
E 100
K
E
q
400
0
0
400
100
200
400
200
283
400
300
346
400
400
400
400
500
447
q  447  400  47
The Total Product and Marginal Product curves
Marginal Product of Labor
Total Product (of Labor)
500
0.30
450
0.25
400
350
300
w
MPL
q
If p = $1 per unit …
0.20
250
200
0.15
0.10
150
100
0.05
LD
50
0.00
0
0
100
200
300
Em ploym ent
400
500
600
0
100
200
300
400
500
Em ploym ent
The total product curve gives the relationship between output and the
number of workers hired by the firm (holding capital fixed).
The marginal product curve gives the output produced by each additional
worker, and the average product curve gives the output per worker.
If we multiply each MPL value by p we get the VPL, the resulting graph is
the firm’s labor demand.
Profit Maximization
• Perfectly competitive firms cannot influence p, w, or r. Suppose p = 200,
w = 70 and r = 30. In the short-run K is constant at say 100.
• The short-run production function is
q  10100
E E
• Fixed capital expenses:
r  K  3, 000
• Variable labor expenses
w  E  70  E
• Total production expenses
3, 000  70  E
Profit Maximization
• Perfectly competitive firms cannot influence p, w, or r. Suppose p =
200, w = 70 and r = 30. In the short-run K is constant at say 100.
• Revenue
p  q  200  q 

p  q  200 10 E

p  q  2000 E
• Short-run profit
profit  2000 E  3000  70  E
Profit Maximization
TE
Rev
Slope Rev = Slope of TE
VMP = w
p ∙ MPL = w
FE
E
The profit max condition:
Slope profit = 0
E *  204
E
profit
Short-run Profit Maximization
• Maximum profits occur when the profit curve reaches its peak (slope = 0)
  2000 E 0.5  3000  E 0  70  E1

 2000 E 0.51 (0.5)  3000 E 01 (0)  70  E11 (1)
E
Profit maximizing
employment
14.29  E
E  204
0.5

 1000 E 0.5  70
E
1000 E 0.5  70  0
Slope of profit
1000
1000E
E 0.5  70
w
VMP = (0.5)(200)(10)E –0.5 = w
Labor demand
equation
Labor Demand Curve
• The demand curve for labor indicates how the firm reacts to wage
changes, holding K = 100, r = 30, and p = 200 constant
w  1000 E 0.5
wage
E
w
2500
20
625
40
204
70
70
40
20
204
625
2500
Employment
Labor Demand Curve
•
Recall VMP = (0.5)(200)(100 0.5)E –0.5 = 1000E –0.5
•
Since p = 200 and K = 100, the most general form of the labor demand curve is
w  1000 E 0.5
wage
w  (0.5)( p)( K 0.5 ) E 0.5
70
40
p
K
E
w
200
100
204
70
250
100
319
70
250
400
1276
70
20
204 319
1276
Employment
Profit Maximization Rules
• The profit maximizing firm should produce up to the point where
the cost of producing an additional unit of output (marginal cost) is
equal to the revenue obtained from selling that output (marginal
revenue)
Choose q* so that
MR = MC
• Marginal Productivity Condition: this is the hiring rule, hire labor
up to the point when the added value of marginal product equals the
added cost of hiring the worker (i.e., the wage)
Choose E* so that
VMP = w
Long-run Production
• In the long run, the firm maximizes profits by choosing how many workers
to hire AND how much plant and equipment to invest in
q  K E
• Isoquant: describes the possible combinations of labor and capital that
produce the same level of output, say at q0 = 500 units.
500  K  E
5002  K  E
• Isoquants…
–
–
–
–
–
K  5002 E 1
Must be downward sloping
Cannot intercept
That are higher indicate more output
Are convex to the origin
slope is the negative ratio of MPK and MPL
Isoquant curves
•
Example: Isoquant curve with q0 = 500
K  5002 E 1
1250
capital
E
K
200
1250
400
625
1200
208
625
208
q0 = 500
200
400
1200
Employment
Isoquant curves
•
Example: Isoquant curve with q1 = 600
K  6002 E 1
capital
E
K
200
1800
400
900
1200
300
900
300
q0 = 600
q0 = 500
200
400
1200
Employment
Isocost lines
• The Isocost line indicates the possible combinations of labor and
capital the firm can hire given a specified budget
C0 = rK + wE
C0 – wE = rK
K
C0 w
 E
r
r
• Isocost indicates equally costly combinations of inputs
• Higher isocost lines indicate higher costs
Isocost lines
•
Example: Suppose w = $70 per hour, r = $30 per hour, and C0 = $45,840.
K
K
C0 w
 E
r
r
45840 70
 E
30
30
K  1528  2.333  E
E
K
200
1061
400
595
600
128
1528
capital
1061
595
128
200
400
Employment
600
C0 = 45840
Isocost lines
•
Example: What happens if costs rise to C1 = $50,400
K
K
C0 w
 E
r
r
50400 70
 E
30
30
K  1680  2.333  E
1680
1528
1213
capital
1061
747
E
K
200
1213
400
747
128
600
280
128
595
C1 = 50400
200
400
Employment
600
C0 = 45840
Isocost lines
•
Whenhappens
r = $30 if r decreases to $27
Example: What
K  1528  2.333  E
45840 70
K

E
27
27
C0 = 70(655) + 30(0) = $45,850
1698
K  1698  2.593  E
C0 = 70(655) + 27(0) = $45,850
1528
capital
1179
1061
E
K
K
200
1061
1179
655
0
0
0
200
655
Employment
Isocost lines
•
Whenhappens
w = $70if r decreases to $27
Example: What
K  1528  2.333  E
45840 55
K
 E
30
30
1528
C0 = 70(0) + 30(1528) = $45,840
C0 = 55(0) + 30(1528) = $45,840
1179
K  1528 1.833 E
1061
E
K
K
200
1061
1179
655
0
327
capital
327
200
655
Employment
Long-run cost minimization
•
Example: Suppose w = $70 per hour, r = $30 per hour, and q0 = 500
E* = 327
K* = 765
q  K E
K
q*  765  327
q*  500
C2 = 70(204) + 30(1200) = 50280
C*1 = 70(327) + 30(765) = 45840
C0 = 70(204) + 30(834) = 39300
E
Long-run cost minimization
•
This least cost choice is where the isocost line is tangent to the isoquant
– i.e., Marginal rate of substitution = w/r
•
Profit maximization implies cost minimization
– The firm produces q0 = 500 units no matter what the K and E are.
– The competitive firm is a price taker not a price maker (p = 91.68 was given)
– Hence firm revenue = $45,840 no matter what the K and E are.
• On the highest isocost line the firm would lose $4440 because C2 = 50280
• On the lowest isocost line the firm is unable to make 500 units
• On the “just right” isocost line the breaks even because C* = $45,840.
Long Run Demand for Labor
• If the wage rate drops, two effects take place
– Firm takes advantage of the lower price of labor by expanding
production (scale effect)
• q can be increased at the same cost!
– Firm takes advantage of the wage change by rearranging its mix of
inputs (while holding output constant; substitution effect)
Long Run Demand for Labor
•
Example: Suppose w falls to 60 per hour
1528
C* = 60(374) + 30(780) = $45,840
E* = 327
K* = 765
q*  500
q*  780  374
780
765
q*  540
capital
p = 91.68
Profit = $3667.20
327 374
Employment
Long Run Demand Curve for Labor
Dollars
When w = 70, E* = 327
When w = 60, E* = 374
70
60
DLR
327
374
Employment
Substitution and Scale Effects
capital
q1
q0
scale
sub
Employment
Elasticity of Substitution
• The curvature of the isoquant measures elasticity of substitution
• Intuitively, elasticity of substitution is the percentage change in capital to
labor (a ratio) given a percentage change in the price ratio (wages to real
interest)
%   KL 
%   wr 
• This is the percentage change in the capital/labor ratio given a 1% change
in the relative price of the inputs (w/r)
Imperfect substitutes in labor
Black Labor
An affirmative action program
can encourage the discriminatory
firm to minimize cost
A discriminatory firm hires fewer
blacks than what is optimal
and hires more whites
(it might have to import them!)
Discriminatory firms production costs
are higher than they would have been
had they been color-blind
q*
White Labor
Imperfect substitutes in labor
Black Labor
An affirmative action program
forces the color-blind firm to
hire
more blacks
An affirmative
action raises the
color-blind
firm’s
production cost
Which means
the color-blind
firm must hire fewer whites
A color-blind firm hires relatively more
whites because of the shape of the isoquants.
q*
White Labor
Other types of isoquants
Capital
Capital
100
q 0 Isoquant
q 0 Isoquant
5
200 Employment
20
Employment
Capital and labor are perfect
substitutes if the isoquant is linear.
The two inputs are perfect complements
if the isoquant is right-angled.
Hence, the firm can substitute two
workers with one machine and not
The firm then gets the same output
when it hires 5 machines and 20
workers as when it hires 5 machines and
see its output change.
25 workers.