• Describes the technology that the firm uses to produce goods and services – E.g., • The more E and K the higher the firm’s output q K E 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
Dec 29, 2015
• Describes the technology that the firm uses to produce goods and services
– E.g.,
• The more E and K the higher the firm’s output
q K E
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
Short-run Production Function
• Over the long-run K varies, but in the short-run K is fixed
– E.g., K = 400 and
• The more E the higher the firm’s short-run output
400q E
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
20q E
Law of diminishing marginal productivity
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
• The marginal product of labor is (MPL) the change in output resulting from hiring an additional worker, holding constant the quantities of other inputs
100 0 100E
2002
100
qMPL
E
200 0 200q
Law of diminishing marginal productivity
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
• The marginal product of labor is (MPL) the change in output resulting from hiring an additional worker, holding constant the quantities of other inputs
200 100 100E
830.83
100
qMPL
E
283 200 83q
Law of diminishing marginal productivity
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
• The marginal product of labor is (MPL) the change in output resulting from hiring an additional worker, holding constant the quantities of other inputs
300 200 100E
630.63
100
qMPL
E
346 283 63q
Law of diminishing marginal productivity
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
• The marginal product of labor is (MPL) the change in output resulting from hiring an additional worker, holding constant the quantities of other inputs
400 300 100E
540.54
100
qMPL
E
400 346 54q
Law of diminishing marginal productivity
K E q
400 0 0
400 100 200
400 200 283
400 300 346
400 400 400
400 500 447
• The marginal product of labor is (MPL) the change in output resulting from hiring an additional worker, holding constant the quantities of other inputs
500 400 100E
470.47
100
qMPL
E
447 400 47q
The Total Product and Marginal Product curves
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.
Total Product (of Labor)
0
50
100
150
200
250
300
350
400
450
500
0 100 200 300 400 500 600
Employment
q
Marginal Product of Labor
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400 500
Employment
MP
L
If p = $1 per unit …
Value
LD
w
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
• Fixed capital expenses:
• Variable labor expenses
• Total production expenses
100q E 10q E
3,000r K
70w E E
3,000 70 E
200 10p q E
• 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
• Short-run profit
2000 3000 70profit E E
200p q q
2000p q E
Profit Maximization
The profit max condition:
FE
* 204E E
E
Profit Maximization
Slope Rev = Slope of TE
VMP = w
p ∙ MPL = w
Slope profit = 0
RevTE
profit
Slope of profit
0.51000E w
VMP = (0.5)(200)(10)E –0.5 = w
0.514.29 E
Short-run Profit Maximization
• Maximum profits occur when the profit curve reaches its peak (slope = 0) 0.5 0 12000 3000 70E E E
0.5 01 1 112000 3000 7(0.5) (0) (1)0E E EE
0.51000 70EE
0.51000 70 0E
0.51000 70E
Labor demand equation
204E
Profit maximizing employment
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
0.51000w E
E w
2500 20
625 40
204 70
70
204 625 2500 Employment
40
20
wage
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
0.51000w E
70
204
40
20
wage
0.5 0.5(0.5)( )( )w p K E
p K E w
200 100 204 70
250 100 319 70
250 400 1276 70319 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
• In the long run, the firm maximizes profits by choosing how many workers to hire AND how much plant and equipment to invest in
• Isoquant: describes the possible combinations of labor and capital that produce the same level of output, say at q0 = 500 units.
• Isoquants…– 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
q K E
500 K E
2500 K E 2 1500K E
Long-run Production
1250
200 400 1200 Employment
625
208
capital
• Example: Isoquant curve with q0 = 500
E K
200 1250
400 625
1200 208
2 1500K E
q0 = 500
Isoquant curves
• Example: Isoquant curve with q1 = 600
E K
200 1800
400 900
1200 300
2 1600K E
900
200 400 1200 Employment
300
capital
q0 = 600
q0 = 500
Isoquant curves
• 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
• Isocost indicates equally costly combinations of inputs
• Higher isocost lines indicate higher costs
0C wK E
r r
Isocost lines
C0 = 45840200 400 600 Employment
• Example: Suppose w = $70 per hour, r = $30 per hour, and C0 = $45,840.
0C wK E
r r
1061
595
128
45840 70
30 30K E
E K
200 1061
400 595
600 128
1528 2.333K E
1528
capital
Isocost lines
C0 = 45840
C1 = 50400
• Example: What happens if costs rise to C1 = $50,400
200 400 600 Employment
0C wK E
r r
1213
747
128
70
30
50 0
30
40K E
capital
E K
200 1213
400 747
600 280
2 31680 . 33K E
1528
1680
1061
595
128
Isocost lines
• Example: When r = $30
200 655 Employment
capital
1528 2.333K E
27
45840
7
0
2
7K E
E K K
200 1061 1179
655 0 0
1698 2.593K E
1061
1179
1528
1698
C0 = 45850
C0 = 45850
C0 = 70(655) + 27(0) = $45,850
C0 = 70(655) + 30(0) = $45,850
• Example: What happens if r decreases to $27
0
Isocost lines
200 655 Employment
capital
1528 2.333K E 45840
3
55
0 30K E
E K K
200 1061 1179
655 0 327
1 81528 . 33K E 1061
1179
1528
327
C0 = 45840
C0 = 45840
C0 = 55(0) + 30(1528) = $45,840
C0 = 70(0) + 30(1528) = $45,840
• Example: When w = $70• Example: What happens if r decreases to $27
Isocost lines
q K E
E* = 327
K* = 765
* 765 327q
• Example: Suppose w = $70 per hour, r = $30 per hour, and q0 = 500
C2
C0
C1
+ 30(1200) = 50280= 70(204)= 70(327)
+ 30(765)
= 45840= 70(204) + 30(834) = 39300
C*
* 500q
E
K
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 cost minimization
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)
327
Employment 374
capital
780
1528
765
C* = 45840
C* = 45840
C* = 60(374) + 30(780) = $45,840
• Example: Suppose w falls to 60 per hour
Long Run Demand for Labor
* 780 374q
* 540q
E* = 327
K* = 765* 500q
p = 91.68
Profit = $3667.20
Long Run Demand Curve for Labor
60
70
Dollars
Employment
DLR
When w = 70, E* = 327
374327
When w = 60, E* = 374
• 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)
• This is the percentage change in the capital/labor ratio given a 1% change in the relative price of the inputs (w/r)
%
%
KL
wr
Elasticity of Substitution
Imperfect substitutes in labor
q*
Discriminatory firms production costs are higher than they would have been had they been color-blind
Black Labor
White Labor
A discriminatory firm hires fewer blacks than what is optimal
and hires more whites (it might have to import them!)
An affirmative action programcan encourage the discriminatory firm to minimize cost
q*
An affirmative action program forces the color-blind firm to hire more blacks
A color-blind firm hires relatively more whites because of the shape of the isoquants.
Black Labor
White Labor
Which means the color-blind firm must hire fewer whites
An affirmative action raises the color-blind firm’s production cost
Imperfect substitutes in labor
Capital
Employment
q 0 Isoquant
100
200
5
20
Capital
Employment
q 0 Isoquant
Other types of isoquants
Capital and labor are perfect substitutes if the isoquant is linear.
Hence, the firm can substitute two workers with one machine and not see its output change.
The two inputs are perfect complements if the isoquant is right-angled.
The firm then gets the same output when it hires 5 machines and 20 workers as when it hires 5 machines and 25 workers.