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Section 3 – Theory of the Firm

•Thus far we have focused on the individual consumer’s decisions:

– Choosing consumption and leisure to:• Maximize Utility• Minimize Income

•Section 3 deals with another economic agent, the producer, and their decisions:

– Choose inputs, production in order to:• Minimize Costs (to hopefully maximize

profits)

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Section 3 – Theory of the Firm

In this section we will cover:

Chapter 6: Inputs and Production Functions

Chapter 7: Costs and Cost Minimization

Chapter 8: Cost Curves

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Chapter 6: Inputs and Production Functions

Consumer Theory

Theory of the Firm

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Chapter 6: Inputs and Production Functions

In this chapter we will cover:

6.1 Inputs and Production6.2 Marginal Product (similar to marginal utility)

6.3 Average Product6.4 Isoquants (similar to indifference curves)

6.5 Marginal rate of technical substitution (MRTS, similar to MRS)

6.6 Special production functions (similar to special utility functions)

6.7 Technological Progress

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Inputs: Productive resources, such as labor and capital, that firms use to manufacture goods and services (also called factors of production)

Output: The amount of goods and services produced by the firm

Production: transforms inputs into outputs

Technology: determines the quantity of output possible for a given set of inputs.

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Production function: tells us the maximum possible output that can be attained by the firm for any given quantity of inputs.

Q = f(L,K,M)Q = f(P,F,L,A)Computer Chips = f1(L,K,M)Econ Mark = f2(Intellect, Study, Bribe)

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Production and Utility Functions

•In Consumer Theory, consumption of GOODS lead to UTILITY:

U=f(kraft dinner, wieners)

•In Production Theory, use of INPUTS causes PRODUCTION:

Q=f(Labour, Capital, Technology)

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A technically efficient firm is attaining the maximum possible output from its inputs (using whatever technology is appropriate)A technically inefficient firm is attaining less than the maximum possible output from its inputs (using whatever technology is appropriate)

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production set : all points on or below the production function

Note: Capital refers to physical capital (goods that are themselves produced goods) and not financial capital (the money required to start or maintain production).

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Example: The Production Function and Technical Efficiency

Q = f(L)

L

Q

•••

C

D

B

Production Set

Inefficient point

Production Function

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Causes of technical inefficiency:

1) Shirking-Workers don’t work as hard as they can-Can be due to laziness or a union strategy

2) Strategic reasons for technical inefficiency-Poor production may get government grants-Low profits may prevent competition

3) Imperfect information on “best practices”-inferior technology

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Acme medical equipment faces the production function:

Q=K1/2L1/2

Given labour of 10 and capital of 20, is Acme producing efficiently by producing 12 units?

What level of production is technically efficient?

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Q =K1/2L1/2

=201/2101/2

=14.14

Acme is not operating efficiently by producing 12 units. Given labour of 10 and capital of 20, Acme should be producing 14.14 units in order to be technically efficient.

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6.2 Marginal Product

• The production function calculates TOTAL PRODUCT

•Marginal Product of an input: the change in output that results from a small change in an input holding the levels of all other inputs constant. MPL = Q/L (holding constant all other inputs)

MPK = Q/K (holding constant all other inputs)

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Marginal Utility and Marginal Product

•In Consumer Theory, marginal utility was the slope of the total utility curve

•In Production Theory, marginal product is the slope of the total product curve:

16L

MPL

Q

LMPL increasing

MPL decreasing

MPL becomes negative

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Law of Diminishing Returns•Law of diminishing marginal utility: marginal utility (eventually) declines as the quantity consumed of a single good increases.

•Law of diminishing marginal returns states that marginal products (eventually) decline as the quantity used of a single input increases.

•Generally the first few inputs are highly productive, but additional units are less productive (ie: computer programmers working in a small room)

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Example: Production as workers increase

L

Q

Each Additional workerIs more productive

Total Product

Each AdditionalworkerIs equallyproductive

Each AdditionalworkerIs lessproductive

Each AdditionalworkerDecreases Production

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Average product: total output that is to be produced divided by the quantity of the input that is used in its production:

APL = Q/L APK = Q/K Example:Q=K1/2L1/2

APL = [K1/2L1/2]/L = (K/L)1/2

APK = [K1/2L1/2]/K = (L/K)1/2

6.3 Average Product

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Marginal, and Average Product

When Marginal Product is greater than average product, average product is increasing

-ie: When you get an assignment mark higher than your average, your average increases

When Marginal Product is less than average product, average product is decreasing

-ie: When you get an assignment mark lower than your average, your average decreases

Therefore Average Product is maximized when it equals marginal product

21L

APL

MPL

Q

LAPL increasing

APL maximized

APL decreasing

APL

MPL

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Isoquant: traces out all the combinations of inputs (labor and capital) that allow that firm to produce the same quantity of output.

Example: Q = 4K1/2L1/2

 

What is the equation of the isoquant for Q = 40?  40 = 4K1/2L1/2

  => 100 = KL => K = 100/L

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…and the isoquant for Q = Q*?  Q* = 4K1/2L1/2

 Q*2 = 16KLK = Q*2/16L

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L

K

Q = 20

Q = 40

All combinations of (L,K) along theisoquant produce 40 units of output.

0

Slope=K/L

Example: Isoquants

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Indifference and Isoquant Curves

•In Consumer Theory, the indifference curve showed combinations of goods giving the same utility•The slope of the indifference curve was the marginal rate of substitution

•In Production Theory, the isoquant curve shows combinations of inputs giving the same product•The slope of the isoquant curve is the marginal rate of technical substitution:

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Marginal rate of technical substitution (labor for capital): measures the amount of K the firm the firm could give up in exchange for an additional L, in order to just be able to produce the same output as before.

Marginal products and the MRTS are related:

MPL/MPK = -K/L = MRTSL,K

6.5 Marginal Rate of Technical Substitution (MRS)

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Marginal Rate of Technical Substitution (MRS)

KL

K

KL

MRTSLK

LK

KL

KMPLMPQ

,K

Lconstantoutput

constantoutput K

L

L

curve,isoquant thealong moves one as 0Q sincebut

MPMP

MPMP

MPMP-

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The marginal rate of technical substitution, MRTSL,K tells us:

The amount capital can be decreased for every increase in labour, holding output constant

ORThe amount capital must be increased for every decrease in labour, holding output constant

-as we move down the isoquant, the slope decreases, decreasing the MRTSL,K

-this is diminishing marginal rate of technical substitution-as you focus more on one input, the other

input becomes more productive

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MRTS ExampleLet Q=4LKMPL=4KMPK=4LFind MRTSL,K

MRTSL,K = MPL/MPK

MRTSL,K =4K/4LMRTSL,K =K/L

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Isoquants – Regions of Production

•Due to the law of diminishing marginal returns, increasing one input will eventually decrease total output (ie: 50 workers in a small room)

•When this occurs, in order to maintain a level of output (stay on the same isoquant), the other input will have to increase

•This type of production is not economical, and results in backward-bending and upward sloping sections of the isoquant:

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Example: The Economic and the Uneconomic Regions of Production

L

K

Q = 10

Q = 20

0

MPK < 0

MPL < 0

IsoquantsUneconomic region

Economic region

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Isoquants and Substitution

•Different industries have different production functions resulting in different substitution possibilities:

– ie: In mowing lawns, hard to substitute away from lawn mowers

•In general, it is easier to substitute away from an input when it is abundant

–This is shown on the isoquant curve

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MRTSL,K is high; labour is scarce so a little more labour frees up a lot of capital

K

L

•

•

MRTSL,K is low; labour is abundant so a little more labour barely affects the need for capital

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MRTS ExampleLet Q=4LKMPL=4KMPK=4LMRTSL,K =K/LShow diminishing MRTS when Q=16.

When Q=16, (L,K)=(1,4), (2,2), (4,1)

MRTS(1,4)=4/1=4MRTS(2,2)=2/2=1MRTS(4,1)=1/4

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MRTSL,K =4

K

L

•

•MRTSL,K =1/4

4

2

1 •1 2 4

MRTSL,K =1

Q=16

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When input substitution is easy, isoquants are nearly straight lines

K

L

When input substitution is hard when inputs are scarce, isoquants are more L-shaped

55 100

170

130

100

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How much will output increase when ALL inputs increase by a particular amount?

RTS = [%Q]/[%(all inputs)]1% increase in inputs => more than 1% increase in

output, increasing returns to scale.

1% increase in inputs => 1% increase in outputconstant returns to scale.

1% increase in inputs => a less than 1% increase in output, decreasing returns to scale.

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Example 1: Q1 = 500L1+400K1

Q1 * = 500(L1)+400(K1)

Q1 *= 500L1+400K1

Q1 *= (500L1+400K1)Q1 *= Q1

So this production function exhibits CONSTANT returns to scale. Ie: if inputs double (=2), outputs double.

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Example 2: Q1 = AL1K

1

Q2 = A(L1)(K1) = + AL1

K1

= +Q1

so returns to scale will depend on the value of +.

+ = 1 … CRS+ <1 … DRS+ >1 … IRS

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Why are returns to scale important?

If an industry faces DECREASING returns to scale, small factories make sense

-It is easier to have small firms in this industry

If an industry faces INCREASING returns to scale, large factories make sense -Large firms have an advantage; natural monopolies

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• The marginal product of a single factor may diminish while the returns to scale do not

• Marginal product deals with a SINGLE input increasing, while returns to scale deals with MULTIPLE inputs increasing

• Returns to scale need not be the same at different levels of production

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1. Linear Production Function: Q = aL + bK

MRTS constantConstant returns to scaleInputs are PERFECT SUBSTITUTES:

-Ie: 10 CD’s are a perfect substitute for 1 DVD for storing data.

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Example: Linear Production Function

L

K

Q = Q0

Q = Q1

0

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-ie: 2 pieces of bread and 1 piece of cheese make a grilled cheese sandwich: Q=min (c, 1/2b)

45Bread

Cheese

2 4

Q = 1

Q = 2

0

1

2

Example: Fixed Proportion Production Function

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3.Cobb-Douglas Production Function: Q = aLK

if + > 1 then IRTS if + = 1 then CRTS if + < 1 then DRTS

smooth isoquants MRTS varies along isoquants

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Example: Cobb-Douglas Production Function

L

K

0

Q = Q1

Q = Q0

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4. Constant Elasticity of Substitution Production Function:

Q = [aL+bK]1/

Where = (-1)/

if = 0, we get Leontief caseif = , we get linear caseif = 1, we get the Cobb-Douglas case

General form of other functions

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Definition: Technological progress shifts isoquants inward by allowing the firm to achieve more output from a given combination of inputs (or the same output with fewer inputs).

Neutral technological progress shifts the isoquant corresponding to a given level of output inwards, but leaves the MRTSL,K unchanged along any ray from the origin

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K/L

MRTS remains same

Q = 100 before

Q = 100 after

Example: “neutral technological progress”K

L

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Labor saving technological progress results in a fall in the MRTSL,K along any ray from the origin

Slope decreases along origin raySince MRTSL,K=MPL/MPK, MPk increases more than MPL

ie: better robots, computers, machines

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Example: Labor Saving Technological Progress

K/L

MRTS gets smaller

Q = 100 before

Q = 100 after

K

L

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Capital saving technological progress results in a rise in the MRTSL,K along any ray from the origin.

Slope increases along origin raySince MRTSL,K=MPL/MPK, MPk increases less than MPL

ie: higher education, higher skilled workers

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K/L

MRTS gets larger

Q = 100 before

Q = 100 after

Example: “capital saving technological progress”K

L

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L2Q :progressTech Saving-Labour

2LQ :progressTech Saving-Capital

L4Q :progressTech Neutral

L2Q :Originally

:Examples

K

K

K

K

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Chapter 6 Key ConceptsInputs and Production

Technical EfficiencyMarginal Product

Law of Diminishing ReturnsAverage ProductIsoquantsMarginal rate of technical substitution

Returns to ScaleSpecial production functionsTechnological Progress