Short-run Production Function

• 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


K E q

400 0 0

400 100 200

400 200 283

400 300 346

400 400 400

400 500 447


200 100 100E

830.83

100

qMPL

E

283 200 83q


K E q

400 0 0

400 100 200

400 200 283

400 300 346

400 400 400

400 500 447


300 200 100E

630.63

100

qMPL

E

346 283 63q


K E q

400 0 0

400 100 200

400 200 283

400 300 346

400 400 400

400 500 447


400 300 100E

540.54

100

qMPL

E

400 346 54q


K E q

400 0 0

400 100 200

400 200 283

400 300 346

400 400 400

400 500 447


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


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


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


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

Substitution and Scale Effects

q1

q0

subscale

Employment

capital

• 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.

Documents

Short-run Production Function