Producer Behavior Copyright © 2013 Worth Publishers, All Rights Reserved Microeconomics Goolsbee/Levitt/ Syverson 1/e 6 Chapter outline 6.1The Basics.

Producer Behavior

Copyright © 2013 Worth Publishers, All Rights Reserved Microeconomics Goolsbee/Levitt/ Syverson 1/e

6Chapter outline

6.1 The Basics of Production

6.2 Production in the Short Run

6.3 Production in the Long Run

6.4 The Firm’s Cost-Minimization Problem

6.5 Returns to Scale

6.6 Technological Change

6.7 The Firm’s Expansion Path and the Total Cost Curve

6.8 Conclusion

6-1

Introduction

Copyright © 2013 Worth Publishers, All Rights Reserved Microeconomics Goolsbee/Levitt/ Syverson 1/e 6-2

6We now turn to the supply side of the supply and demand model

•How do firms decide whether and how much to produce?

•How do firms choose between inputs such as capital and labor?

•How does the timeframe of analysis affect firm decisions?

These questions are fundamental to understanding how supply responds to changing market conditions



Production describes the process by which an entity turns raw inputs into a good or service•Final goods are purchased by consumers (e.g., bread)•Intermediate goods are used as inputs in other production processes (e.g., wheat used to produce bread)

Start with a production function•Similar to a utility function for consumers, except more tangible•Mathematical relationship between amount of output and various combinations of inputs

6



Simplifying Assumptions about Firms’ Production Behavior

1.The firm produces a single good

2.The firm has already chosen which product to produce

3.Firms minimize costs associated with every level of production• Necessary condition for profit maximization

4.Only two inputs are used in production: capital and labor• Capital: buildings, equipment, etc.• Labor: All human resources

5.In the short run, firms can choose the amount of labor employed, but capital is assumed to be fixed in total supply

6



6. Output increases with inputs

7. Inputs are characterized by diminishing returns• If the amount of capital is held constant, each additional

worker produces less incremental output than the last, and vice versa

8. The firm can employ unlimited capital and labor at fixed prices, and

9. Capital markets are well functioning (the firm is not budget-constrained)

6



6

LKfQ ,

LKQ

Application


The Environment as an Input to Production

The environment provides many goods and services that are not exchanged in markets•Mangroves serve as a key habitat for juvenile grouper fish, an important commercial species in many parts of the world•This ecosystem service (habitat) is difficult to quantify•A lack of knowledge of values often means mangroves and other ecosystems are destroyed to make way for more salient economic benefits (e.g., beach resort or shrimp farm)

Barbier (2007) provides an overview and application of a methodology for valuing the environment as an input

6

Citation: Barbier, E.B. 2007. “Valuing ecosystem services as productive inputs” Economic Policy January, 177-229.

Images: FreeDigitalPhotos.net

Application


Consider the following production function for a fishery

where h is fishery-wide harvest, the E ’s are “traditional” inputs (# of boats, hours spent fishing, size of net, etc.) and S is the size of the adjacent wetland

More wetlands are associated with lower costs and more aggregate production in an open-access fishery•Model is applied to mangrove wetlands in Thailand•Adjacent to valuable artisanal fisheries

Value of mangroves is found to be $10-12,000 per hectare per year•Includes other values (storm protection, forest products)•Dwarfs $1,200 benefit of conversion for shrimp farming

6

Citation: Barbier, E.B. 2007. “Valuing ecosystem services as productive inputs” Economic Policy January, 177-229. http://gesd.free.fr/bw174.pdf


SEEfh ki ,,...,

6.2


The “short run” refers to the case in which the level of capital is fixed

First, consider how production changes as we vary the amount of labor

Marginal product refers to the additional output that a firm can produce using an additional unit of an input•Similar to marginal utility•Generally assumed to fall as more of an input is used

6

LKfQ ,

Production in the Short Run

6.2


The marginal product of labor, MPL , is given as

Consider the production function

where capital is fixed at four units

Table 6.1 calculates the marginal product of labor for this production function

6

L

QMPL

5.05.0 LKQ

5.05.05.0 24 LLQ


6.2


6Production in the Short Run

6.2



Figure 6.1 A Short-Run Production Function

6.2


Table 6.1 and Figure 6.1 reveal the common assumption of diminishing marginal product associated with production inputs

As a firm employs more of one input, while holding all others fixed, the marginal product of that input will fall

This is seen most easily using a graph


6.2


6Figure 6.2 Deriving the Marginal Product of Labor


(a) (b)MPL(∆ Q /∆ L )5 1.54.4743.46 1.02.832 0.5 MPL

0 0Labor (L) Labor (L)1 2 3 4 5 1 2 3 4 5

Output (Q ) ProductionfunctionSlope = 0.45Slope = 0.5Slope = 0.58Slope = 0.71 1Slope = 1 Slope =L0.5

6.2


Returning to the mathematical representation of MPL ,

and using the example production function

As becomes very small, we use calculus to arrive at the equation for MPL

6

L

LKfLLKf

L

QMPL

,,

5.05.05.0 24 LLQ

L

LLLMPL

5.05.0 22

L

5.05.0

1, LLdL

LKdfMPL


6.2


Another important production metric is average product•Total output divided by the total amount of an input used•The average product of labor is give by

What is the difference between marginal and average product?


L

QAPL


Figure it outProduction of a Bakery

Short-run production function for a local bakery making loaves of bread

Q is the number of loaves produced per hour, is the number of ovens (fixed at 2), and L is the number of workers

Answer the following

1.Write an equation for the short-run production function with output as a function of labor only

2.Calculate total output per hour for

3.Calculate MPL and APL for the same labor levels above

25.075.020, LKLKfQ

K

5 ,4 ,3 ,2 ,1 ,0L



1. Substitute into the production function

2. To calculate total output, simply substitute the different labor quantities into the production function above

2K

25.025.075.0 64.33220 LLQ

Labor Production

L = 0 Q = 33.64(0)0.25 = 0

L = 1 Q = 33.64(1)0.25 = 33.64

L = 2 Q = 33.64(2)0.25 = 40

L = 3 Q = 33.64(3)0.25 = 44.27

L = 4 Q = 33.64(4)0.25 = 47.57

L = 5 Q = 33.64(5)0.25 = 50.30



3. The marginal product of labor is the additional amount of bread produced with one more unit of labor

Average product is simply total output divided by total labor

Are there diminishing returns to labor? How do you know?

Labor Production

L = 0 Q = 33.64(0)0.25 = 0

L = 1 Q = 33.64(1)0.25 = 33.64

L = 2 Q = 33.64(2)0.25 = 40

L = 3 Q = 33.64(3)0.25 = 44.27

L = 4 Q = 33.64(4)0.25 = 47.57

L = 5 Q = 33.64(5)0.25 = 50.30

MPL

—

33.64

6.36

4.27

3.30

2.73

APL

—

33.64

20

14.76

11.89

10.06

6.3


• For our purposes, the long run is defined as a period of time long enough to allow firms to adjust the amount of every input used in production

• Table 6.2 describes a long-run production function in which two inputs, capital and labor, are used to produce various quantities of a product

• Columns represent different quantities of labor• Rows represent different quantities of capital• Each cell in the table shows the quantity of output produced

with the labor and capital represented by the column and row values

6Production in the Long Run

6.3


6Production in the Long Run

6.4


The third assumption about production behavior: firms minimize the cost of production

Cost minimization refers to the firm’s goal of producing a specific quantity of output at minimum cost•This is an example of constrained optimization •The firm will minimize costs subject to a specific amount of output that must be produced

The cost minimization model requires two concepts, isoquants and isocost lines

6The Firm’s Cost-Minimization Problem

6.4


An isoquant is a curve representing combinations of inputs that allow a firm to make a particular quantity of output•Similar to indifference curves from consumer theory


6.4


6Figure 6.3 Isoquants

The Firm’s Cost-Minimization Problem

Capital (K )4

Output, Q = 42

Output, Q = 21 Output, Q = 10 Labor (L )1 2 4

6.4


An isoquant is a curve representing combinations of inputs that allow a firm to make a particular quantity of output•Similar to indifference curves from consumer theory

The slope of an isoquant describes how inputs may be substituted to produce a fixed level of output

This relationship is referred to as the marginal rate of technical substitution: the rate at which the firm can trade input X for input Y, holding output constant (MRTSXY )


6.4


6Figure 6.4 The Marginal Rate of Technical

Substitution


Capital (K )A

B Q =2Labor (L)

The marginal product oflabor is low relative to themarginal product of capital.

The marginal product oflabor is high relative to themarginal product of capital .

6.4


Mathematically, MRTSLK can be derived from the condition that, along an isoquant, quantity of output produced is held constant

Rearranging to find the slope of the isoquant yields the MRTSLK

Moving down an isoquant, the amount of capital used declines•MRTSLK describes the rate at which labor must be substituted for capital to hold the quantity produced constant•As you move down an isoquant, the slope gets smaller, meaning the firm has less capital and each unit is relatively more productive


0 KMPLMPQ KL

K

LLKLK MP

MP

L

KMRTSLMPKMP

6.4

6-28Copyright © 2013 Worth Publishers, All Rights Reserved Microeconomics Goolsbee/Levitt/ Syverson 1/e


6.4


Figure 6.5 The Shape of Isoquants Indicates the Substitutability of Inputs


6.4


The Curvature of Isoquants: Substitutes and ComplementsTo illustrate, consider extreme cases•When inputs are perfect substitutes, they can be traded off in a constant ratio in a production process (MRTS is constant)



6Perfect substitutes

0 Natural gas (‘000 cubic feet)

Oil (barrels) Consider production of electricity from either oil or natural gas (Q is kw-hours)

Assume the plant may switch between fuel sources at a relatively constant rate

MRTS is constant

**Numbers are examples1

6.4

Q = 1

234

2 864 Q = 4Q = 3Q = 2


**Numbers are examples

6.4


The Curvature of Isoquants: Substitutes and Complements

The shape of an isoquant reveals information about the relationship between inputs to production•Relatively straight isoquants imply that the inputs are relatively substitutable•Isoquants with significant curvature imply strong complementarity

To illustrate, consider extreme cases•When inputs are perfect substitutes, they can be traded off at a constant rate as part of a production process (constant MRTS)•When inputs are perfect complements, they must be used in a fixed ratio as part of a production process



6Perfect complements

0 Bus drivers

Buses

1

6.4

Q = 123

1 32

Q = 3Q = 2A B

C


6.4


Isoquant maps show how quantities of inputs are related to output produced

An isocost line shows all of the input combinations that yield the same cost•Similar to the budget constraint facing consumers, equation given by

where C is total cost, R is the “rental rate” of capital, and W is the wage rate

•Rearranging yields capital as a function of the rental rate, wage rate, and labor

•Or, graphically


WLRKC

LR

W

R

CK

6.4


6Figure 6.7 Isocost Lines


Capital (K )54321 C = $50 C = $80 C = $ 1000 Labor (L )1 2 3 4 5 6 7 8 9 10

6.4



6.4


6Figure 6.10 Cost Minimization


Capital (K )B (cost-minimizingcombination)

CA A Q = QCC CB Labor (L )

CC cannot produce Q. CA can produce Q butis more expensive.

6.4


Identifying Minimum Cost: Combining Isoquants and Isocost Lines

Mathematically, tangency occurs where the slope of the isocost line is equal to the slope of the isoquant, or

What does this condition imply?•Costs are minimized when the marginal product per dollar spent is equalized across inputs

QUESTION: What if or


L K L

K

MP MP MPW

R MP R W

K LMP MP

R W K LMP MP

R W


figure it outCost minimization

A firm is employing 25 workers (W = $10/hour) and 5 units of capital (R = $20/hour). At these levels, the marginal product of labor is 25 and the marginal product of capital is 30.

Answer the following

1.Is this firm minimizing costs?

2.If not, what changes should they make?

3.How does the answer to (2) depend on the timeframe of analysis?


figure it outCost minimization

1.Cost minimization occurs when

For this firm, we have

Since these two ratios are not equal, the firm is not minimizing costs

2.As , changing the mix of capital and labor can lead to a lower cost of producing the same quantity of output•The wages to labor and capital are fixed, so to equate these two, the quantity of labor employed must rise and/or the quantity of capital employed must fall•This will shift and/or pivot the isoquant

3.Generally, the short run implies that only the amount of labor employed can be altered

K LMP MP

R W

301.5

2025

2.510

K

L

MP

RMP

W

/ /L KMP w MP r

Application


The Cost of Labor and Automation

Stringent labor laws, the threat of labor strikes, and high payroll taxes in France have made labor more expensive than in many other western countries•3300 page labor code•39% payroll taxes (in U.S., employers pay <10%)

The response has been a push for automation in service delivery (i.e., substitution of capital for labor)•Self-checkout registers at supermarkets•Automated ordering at fast food restaurants•Driverless trains

What is the effect of increasing payroll taxes (a tax on labor) on the choice between labor and capital?

6

Citation: “France and automation: Driverless, workless.” The Economist, November 26, 2011.


Application


6

Citation: “France and automation: Driverless, workless.” The Economist, November 26, 2011.

0 4 Cashiers

Auto checkout computers20 As the relative price of labor increases, the isocost curve pivots inward (C′ )To maintain the same level of production, total costs rise until the new isocost line (C2) is tangent to the old isoquant (point B )A

B

8 12 16 20

161284

2Q = 500

C1C2

10

Consider a supermarket deciding between auto checkout computers and human cashiers

Before the tax, the supermarket was able to serve 500 customers per hour with 8 machines and 8 people (point A )

C′


figure it outIsocost Lines

Suppose the wage rate W = $10/hour and the rental rate of capital R = $20/hour.

1.Write an equation for the isocost line for the firm

2.Draw a graph (with capital on the horizontal axis) showing the isocost line for C = $400. Indicate the intercepts and the slope.

3.Suppose the price of labor increases to $20 per hour. Show what happens to the isocost line on your graph.


figure it outIsocost Lines

1.The isocost line has the form:

For this firm, we have

WLRKC

KLC 2010


Figure it outIsocost Lines

0 Hours of capital

Hours of labor40Slope = −0.5

8 12 16 20

32241682

C1C2

20 Slope = −1

The vertical intercept is equal to $400/w ; the horizontal to $400/r . The slope of the isocost line is equal to −w/r An increase in the wage paid to labor reduces the number of hours of labor that can be purchased with $400



Returns to scale refers to the change in output when all inputs are increased or decreased in the same proportion

Returning to the Cobb-Douglas production function

If we assume , then

If K = 2 and L = 2, then Q = 2What happens if the amount of capital and labor used both double?

6

LKQ 5.0

5.05.0 LKQ

42244 5.05.0 Q



This relationship, whereby production increases proportionally with inputs, is called constant returns to scale

Increasing returns to scale describes production for which changing all inputs by the same proportion changes output more than proportionally

Decreasing returns to scale describes production for which changing all inputs by the same proportion changes output less than proportionally

6



QUESTION: Why might a firm experience increasing returns to scale?

• Fixed costs (e.g., webpage management, advertising contracts) do not vary with output

• Learning by doing may occur, whereby a firm develops more efficient processes as it expands

Generally, firms should not experience decreasing returns to scale• When this phenomenon is observed in data, it often results

from not accounting for all inputs (or attributes); for instance, second manager may not be as competent as first

QUESTION: Are there any examples of true decreasing returns?• Regulatory burden: as firms grow larger, often subject to more

regulations; compliance costs may be significant

6

6.5


6Figure 6.12 Returns to Scale

Returns to Scale

(a) (b) (c)Constant Returns to Scale Increasing Returns to Scale Decreasing Returns to ScaleCapital Capital Capital(K) (K) (K)

4 4 4Q =4 Q =6 Q =32 2 2Q =2 Q = 2.5 Q = 1.81 1 1Q =1 Q =1 Q =10 0 01 2 4 1 2 4 1 2 4Labor (L) Labor (L) Labor (L)

Note: labor and capital doubled between isoquants.


figure it outReturns to Scale

For each of the following production functions, determine if they exhibit constant, decreasing, or increasing returns to scale

a.

b.

c.

LKQ 53

LKQ 5 ,6min3.06.018 LKQ



The simplest way to determine returns to scale is to plug in values for labor and capital, calculate output, then double the inputs and calculate output again

a. Consider K = L = 2Now, double the inputs

Since output doubled when inputs doubled, we have constant returns to scale

1610653 LKQ

32454353 LKQ



b. Consider K = L = 2 again

Now, double the inputs

And once again, we have constant returns to scale

c. K = L = 2Now, double the inputs

Since the new output is less than twice the old output, we have decreasing returns to scale

105 ,6min LKQ

2045 ,46min Q

59.3318 3.06.0 LKQ

68.624418 3.06.0 Q



Examining firm-level production data over time reveals increasing output, even when input levels are held constant•The only way to explain this is by assuming some change to the production function

This change is referred to as total factor productivity growth•An improvement in technology that changes the firm’s production function such that more output is obtained from the same amount of inputs

Often assumed to enter multiplicatively with production

where A is the level of total factor productivity

6

LKAfQ ,



6

0 Labor

CapitalWith old technology, the production of Q* requires L1 labor and K1 capital

When technology improves, the isoquant associated with Q* shifts inward, requiring less labor and capital (L2 and K2 )

ABK1 Q* (old tech)

C1C2

K2

L1Q* (new tech)

L2

6.7


So far, we have only focused on how firms minimize costs, subject to a fixed quantity of output•We can use the cost minimization approach to describe how capital and labor change as output increases

An expansion path is a curve that illustrates how the optimal mix of inputs varies with total output

This allows construction of the total cost curve, which shows a firm’s cost of producing particular quantities

6The Firm’s Expansion Path and Total Cost Curve

6.7


6

0 Labor

Capital

A BQ = 20

C = 100

The Firm’s Expansion Path and Total Cost Curve

0 Output

Total Cost

BA$100Q = 10C = 150 C = 250

CQ = 30

Expansion path$150$250

10 20 30

CTotal cost (TC)

Figure 6.15 The Expansion Path and the Total Cost Curve

6.8 Conclusion


This chapter looked closely at how firms make decisions•Firms are assumed to minimize costs at every level of production•The cost-minimizing combination of inputs occurs where the marginal rate of technical substitution is equal to the slope of the isocost line

In Chapter 7 we delve deeper into the different costs facing firms, and how they change with the level of production

6

Producer Behavior Copyright © 2013 Worth Publishers, All Rights Reserved Microeconomics Goolsbee/Levitt/ Syverson 1/e 6 Chapter outline 6.1The Basics.

Documents

worth publishers

basics of productioncopyright

production functionsimilar

production processes

firm decisions

basics of production6

raw inputs

unlimited capital