Slide 12005 South-Western Publishing

Production Economics

Chapter 6

• Managers must decide not only what to produce for the market, but also how to produce it in the most efficient or least cost manner.

• Economics offers widely accepted tools for judging whether the production choices are least cost.

• A production function relates the most that can be produced from a given set of inputs. » Production functions allow measures of the marginal

product of each input.

Slide 2

• A Production Function is the maximum quantity from any amounts of inputs

• If L is labor and K is capital, one popular functional form is known as the Cobb-Douglas Production Function

• Q = • K • L is a Cobb-Douglas Production Function

• The number of inputs is often large. But economists simplify by suggesting some, like materials or labor, is variable, whereas plant and equipment is fairly fixed in the short run.

The Production Function

Slide 3

• Short Run Production Functions:» Max output, from a n y set of inputs

» Q = f ( X1, X2, X3, X4, X5 ... )

FIXED IN SR VARIABLE IN SR _

Q = f ( K, L) for two input case, where K as Fixed

• A Production Function is has only one variable input, labor, is easily analyzed. The one variable input is labor, L.

The Short Run Production Function

Slide 4

• Average Product = Q / L » output per labor

• Marginal Product =Q / L = dQ / dL» output attributable to last unit of labor applied

• Similar to profit functions, the Peak of MP occurs before the Peak of average product

• When MP = AP, we’re at the peak of the AP curve

Slide 5

Elasticities of Production• The production elasticity of labor,

» EL = MPL / APL = (Q/L) / (Q/L) = (Q/L)·(L/Q)

» The production elasticity of capital has the identical in form, except K appears in place of L.

• When MPL > APL, then the labor elasticity, EL > 1. » A 1 percent increase in labor will increase output by more than 1

percent.

• When MPL < APL, then the labor elasticity, EL < 1. » A 1 percent increase in labor will increase output by less than 1

percent.

Slide 6

Short Run Production Function Numerical Example

L Q MP AP 0 0 --- --- 1 20 20 20 2 46 26 23 3 4 5

70 92 110

24 22 18

23.33 23 22

Marginal Product

L 1 2 3 4 5

Average Product

Labor Elasticity is greater then one,for labor use up through L = 3 units

Slide 7

• When MP > AP, then AP is RISING» IF YOUR MARGINAL GRADE IN THIS CLASS IS HIGHER

THAN YOUR GRADE POINT AVERAGE, THEN YOUR G.P.A. IS RISING

• When MP < AP, then AP is FALLING» IF YOUR MARGINAL BATTING AVERAGE IS LESS THAN

THAT OF THE NEW YORK YANKEES, YOUR ADDITION TO THE TEAM WOULD LOWER THE YANKEE’S TEAM BATTING AVERAGE

• When MP = AP, then AP is at its MAX» IF THE NEW HIRE IS JUST AS EFFICIENT AS THE

AVERAGE EMPLOYEE, THEN AVERAGE PRODUCTIVITY DOESN’T CHANGE

Slide 8

Law of Diminishing Returns

INCREASES IN ONE FACTOR OF PRODUCTION, HOLDING ONE OR OTHER FACTORS FIXED, AFTER SOME POINT, MARGINAL PRODUCT DIMINISHES.

A SHORT RUN LAW

point ofdiminishing

returns

Variable input

MP

Slide 9

Figure 7.4 on Page 306

Three stages of production

• Stage 1: average product rising.

• Stage 2: average product declining (but marginal product positive).

• Stage 3: marginal product is negative, or total product is declining.

L

Total Output

Stage 1

Stage 2

Stage 3

Slide 10

Optimal Use of the Variable Input

• HIRE, IF GET MORE REVENUE THAN COST

• HIRE if

TR/L > TC/L• HIRE if the marginal

revenue product > marginal factor cost: MRP L > MFC L

• AT OPTIMUM,

MRP L = W MFC

MRP L MP L • P Q = W

optimal labor

MPL

MRPL

W W MFC

L

wage

•

Slide 11

MRP L is the Demand for Labor

• If Labor is MORE productive, demand for labor increases

• If Labor is LESS productive, demand for labor decreases

• Suppose an EARTHQUAKEEARTHQUAKE destroys capital

• MP L declines with less capital, wages and labor are HURT

D L

D’ L

S L

W

L’ L

Slide 12

Long Run Production Functions• All inputs are variable

» greatest output from any set of inputs

• Q = f( K, L ) is two input example

• MP of capital and MP of labor are the derivatives of the production function» MPL = Q /L = Q /L

• MP of labor declines as more labor is applied. Also the MP of capital declines as more capital is applied.

Slide 13

Isoquants & LR Production Functions

• In the LONG RUN, ALL factors are variable

• Q = f ( K, L )

• ISOQUANTS -- locus of input combinations which produces the same output (A & B or on the same isoquant)

• SLOPE of ISOQUANT is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution

ISOQUANT MAP

B

A

C

Q1

Q2

Q3

K

L

Slide 14

Optimal Combination of Inputs• The objective is to minimize

cost for a given output• ISOCOST lines are the

combination of inputs for a given cost, C0

• C0 = CL·L + CK·K • K = C0/CK - (CL/CK)·L• Optimal where:

» MPL/MPK = CL/CK·» Rearranged, this becomes the

equimarginal criterion

Equimarginal Criterion: Produce where MPL/CL = MPK/CK where marginal products per dollar are equal

Figure 7.9 on page 316

Q(1)

D

L

K

at D, slope of isocost = slope of isoquant

C(1)

Slide 15

Use of the Equimarginal Criterion• Q: Is the following firm

EFFICIENT?

• Suppose that:» MP L = 30

» MPK = 50

» W = 10 (cost of labor)

» R = 25 (cost of capital)• Labor: 30/10 = 3• Capital: 50/25 = 2

• A: No!

• A dollar spent on labor produces 3, and a dollar spent on capital produces 2.

USE RELATIVELY MORE LABOR!

• If spend $1 less in capital, output falls 2 units, but rises 3 units when spent on labor

• Shift to more labor until the equimarginal condition holds.

• That is peak efficiency.

Slide 16

Allocative & Technical Efficiency

• Allocative Efficiency – asks if the firm using the least cost combination of input» It satisfies: MPL/CL = MPK/CK

• Technical Efficiency – asks if the firm is maximizing potential output from a given set of inputs» When a firm produces at point T

rather than point D on a lower isoquant, they firm is not producing as much as is technically possible. Q(1)

D

Q(0)T

Slide 17

Returns to Scale• A function is homogeneous of degree-n

» if multiplying all inputs by (lambda) increases the dependent variable byn

» Q = f ( K, L)

» So, f(K, L) = n • Q

• Constant Returns to Scale is homogeneous of degree 1. » 10% more all inputs leads to 10% more output.

• Cobb-Douglas Production Functions are homogeneous of

degree +

Slide 18

Cobb-Douglas Production Functions

• Q = A • K • L is a Cobb-Douglas Production Function

• IMPLIES:

» Can be CRS, DRS, or IRS

if + 1, then constant returns to scale

if + < 1, then decreasing returns to scale

if + > 1, then increasing returns to scale

• Coefficients are elasticities is the capital elasticity of output, often about .67

is the labor elasticity of output, often about .33which are EK and E L

Most firms have some slight increasing returns to scale

Slide 19

ProblemSuppose: Q = 1.4 L .70 K .35

1. Is this function homogeneous?

2. Is the production function constant returns to scale?

3. What is the production elasticity of labor?

4. What is the production elasticity of capital?

5. What happens to Q, if L increases 3% and capital is cut 10%?

Slide 20

Answers

1. Yes. Increasing all inputs by , increases output by 1.05. It is homogeneous of degree 1.05.

2. No, it is not constant returns to scale. It is increasing Returns to Scale, since 1.05 > 1.

3. .70 is the production elasticity of labor

4. .35 is the production elasticity of capital

5. %Q = EL• %L+ EK • %K = .7(+3%) + .35(-10%) = 2.1% -3.5% = -1.4%