production theory and estimation chapter 7. managers must decide not only what to produce for the...

Production Theoryand Estimation

Chapter 7

• 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.

Green Power Initiatives

California permitted and encouraged buying cheap power from other states.

So, PG&E and Southern Cal Edison scaled back their expansion of production facilities.

Off-Peak and Peak costs per MWh ranged from $25 to over $65, but regulators tried to keep prices low. Resulting in PG&E bankruptcy

Green Power Initiatives Carbon dioxide emission trading schemes

in Europe encouraged construction of greener nuclear & wind generation.

Q: If you were asked to pay 3 times more for electricity in the day than night, would you change your usage?

The Organization of Production

Inputs Labor, Capital, Land

Fixed Inputs Variable Inputs Short Run

At least one input is fixed Long Run

All inputs are variable

Production Function With Two Inputs

K Q6 10 24 31 36 40 395 12 28 36 40 42 404 12 28 36 40 40 363 10 23 33 36 36 332 7 18 28 30 30 281 3 8 12 14 14 12

1 2 3 4 5 6 L

Q = f(L, K)

Production Function With One Variable Input

Total Product

Marginal Product

Average Product

Production orOutput Elasticity

TP = Q = f(L)

MPL =TP L

APL =TP L

EL =MPL

APL


L Q MPL APL EL

0 0 - - -1 3 3 3 12 8 5 4 1.253 12 4 4 14 14 2 3.5 0.575 14 0 2.8 06 12 -2 2 -1

Total, Marginal, and Average Product of Labor, and Output Elasticity

Optimal Use of the Variable Input

Marginal RevenueProduct of Labor MRPL = (MPL)(MR)

Marginal ResourceCost of Labor MRCL =

TC L

Optimal Use of Labor MRPL = MRCL


L MPL MR = P MRPL MRCL

2.50 4 $10 $40 $203.00 3 10 30 203.50 2 10 20 204.00 1 10 10 204.50 0 10 0 20

Use of Labor is Optimal When L = 3.50

Production With Two Variable Inputs

Isoquants show combinations of two inputs that can produce the same level of output.

Firms will only use combinations of two inputs that are in the economic region of production, which is defined by the portion of each isoquant that is negatively sloped.


Isoquants


Economic Region of Production


Marginal Rate of Technical Substitution

MRTS = -K/L = MPL/MPK


MRTS = -(-2.5/1) = 2.5


Perfect Substitutes Perfect Complements

Optimal Combination of Inputs

Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

C wL rK

C wK L

r r

C Total Cost

( )w Wage Rateof Labor L

( )r Cost of Capital K


Prepared by Robert F. Brooker, Ph.D. Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved. Slide 21

Isocost Lines

AB C = $100, w = r = $10

A’B’ C = $140, w = r = $10

A’’B’’ C = $80, w = r = $10

AB* C = $100, w = $5, r = $10



MRTS = w/r


Effect of a Change in Input Prices

Returns to Scale

Production Function Q = f(L, K)

Q = f(hL, hK)

If = h, then f has constant returns to scale.

If > h, then f has increasing returns to scale.

If < h, the f has decreasing returns to scale.

Returns to Scale


Constant Returns to Scale

Increasing Returns to Scale

Decreasing Returns to Scale

Empirical Production Functions

Cobb-Douglas Production Function

Q = AKaLb

Estimated using Natural Logarithms

ln Q = ln A + a ln K + b ln L

The Production Function

A Production Function is the maximum feasible 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 typically greater than

just K & L. 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 (con’t)

The Short Run Production Function Short Run Production Functions:

Max output, from a n y set of inputsQ = f ( X1, X2, X3, X4, X5 ... )

FIXED IN SR VARIABLE IN SR _ _

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

© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a

publicly accessible website, in whole or in part.

The Short Run Production Function (con’t)

A Production Function with only one variable input is easily analyzed. The one variable input is labor, L. Q = f( L )

Average Product = Q / L output per labor

Marginal Product = Q/L =Q/L = dQ/dLoutput 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, this is the peak of the AP curve

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 of

diminishingreturns

Variable input

MP

Bottlenecks in Production Plants

Boeing found diminishing returns in ramping up production.

It sought ways to adopt lean production techniques, cut order sizes, and outsourced work at bottlenecked plants.

Increasing Returns and Network Effects

There are exceptions to the law of diminishing returns.

When the installed base of a network product makes efforts to acquire new customers increasing more productive, we have network effects Outlook and Microsoft Office

Table 7.2: Total, Marginal & Average Products

Total, Marginal & Average Products

Marginal Product

3 4 5 6 7 8

Average Product

The maximum MP occurs before the maximum AP

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

Three stages of production

Three stages of production

Stage 1: average product rising. Increasing returns

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

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

Determining the 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

•

Production Functions with multiple variable inputs

Suppose several 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.

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 Points A & B are on the same

isoquant SLOPE of ISOQUANT from A to

B is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution = -K /L

ISOQUANT MAP

B

A

C

Q1

Q2

Q3

K

L

• 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

at D, slope of isocost = slope of isoquant

• 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!

Use of the Equimarginal Criterion

Use of the Equimarginal Criterion

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.

Production Processes and Process Rays under Fixed Proportions

If a firm has five computers and just one person, typically only one computer is used at a time. You really need five people to work on the five computers.

The isoquants for processes with fixed proportions are L-shaped. Small changes in the prices of input may lead to no change in the process.

M is the process ray of one worker and one machine

people

computers1 2 3 4 5 6 7 8 9

5

4

3

2

1

M

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, that firm is not producing as much as is technically possible. Q(1)

D

Q(0)T

Overall Production Efficiency

Suppose a plant produces 93% of what the technical efficient plant (the benchmark) would produce.

Suppose a plant produces 85.7% of what an allocatively efficient plant would produce, due to a misaligning the input mix.

Overall Production Efficiency = (technical efficiency)*(allocative efficiency)

In this case: overall production efficiency = (.93)(.857) = 0.79701 or about 79.7%.

Returns to Scale

If multiplying all inputs by (lambda) increases the dependent variable bythe firm has constant returns to scale (CRS). Q = f ( K, L) So, f(K, L) = • Q is Constant Returns to Scale So if 10% more all inputs leads to 10% more output the firm

is constant returns to scale. Cobb-Douglas Production Functions are constant returns if

+

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

Suppose: Q = 1.4 K .35 L .70

Is this production function constant returns to scale?

No, it is Increasing Returns to Scale, because 1.05 > 1.

Reasons for Increasing & Decreasing Returns to Scale

Some Reasons for IRS The advantage of

specialization in capital and labor – become more adept at a task

Engineering size and volume effects – doubling the size of motor more than doubles its power

Network effects Pecuniary advantages of

buying in bulk

Some Reasons for DRS Problems with coordination

and control – as a organization gets larger, harder to get everyone to work together

Shirking increases Bottlenecks appear – a form of

the law of diminishing returns appears

CEO can’t oversee a gigantically complex operation

Interpreting the Exponents of the Cobb-Douglas Production Functions

The exponents and are elasticities

is the capital elasticity of output The is [% change in Q / % change in K]

is the labor elasticity of output The is a [% change in Q / % change in L]

These elasticities can be written as EK and E L

Most firms have some slight increasing returns to scale.

Empirical Production ElasticitiesTable 7.4: Most are statistically close to CRS or have IRS

˟ such as management or other staff personnel.

production theory and estimation chapter 7. managers must decide not only what to produce for the...

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