Production

Production Technology 1

Production

So far: We have studied consumers’ behavior.

• Preferences and budgets give us an individual’s choices

• Individual’s choices at a variety of prices yield one

person’s demand

• Add together individual demand to get market demand.

Next: Apply similar ideas to firms. Again, constrained

optimization yields choices, and choices yield supply.

Later: Bring together supply and demand to describe an

entire market.


What do firms choose and why?

Objective: Maximize profit

Choices:

• Outputs

– What goods and services to produce

– Quantities of each output

• Inputs

– Labor (High-skill, low-skill)

– Capital (Machines, computers, buildings…)

– Other (Raw materials, land…)


What do firms choose and why?

Constraints:

• Technology

• Budget (-like) constraints from input and output markets.


Example: General Electric

GE’s Outputs: Appliances, TVs, lighting, insurance, energy,

healthcare, etc.

What markets to pursue and at what scale?

GE’s Inputs: Engineers, factory labor, robots, computers,

glass, metals, etc.

Which inputs to select and in what combinations?

GE’s Constraints: How many ways to make a TV, turbine,

or lightbulb? Do competitors affect GE’s pricing?


Our PlanFirm optimization can be very complicated. Our plan:

• Simplify where possible. (Choose between 2 inputs rather

than among 100s)

• Consider optimization in steps

How will we break the problem into parts?

• Objective: Maximize profit

Profit = Revenue – Costs

Revenue = Output price Output quantity (linear prices)

Cost = Input prices Input quantities

First we consider input quantities, and then we work on output

quantity.


Our Plan1. Describe production technology

– Production functions & substitution between inputs

– Economies of scale

– Short and long run choices

2. Cost-minimization

– Hold fixed a goal for output quantity. What is the best

(cheapest) way to combine inputs to achieve this goal?

– We will cover:

• Appropriate measurement of costs

• Short run v. long run cost minimization

• Cost functions

3. Choose an output quantity


An Aside on Profit Maximization

Is profit maximization realistic?

• Profit maximization is a reliable, reasonable assumption

for many situations. Think of it as a great place to start.

What is going on when firms appear to be doing something

different?

• It’s possible that there other goals

– Intentional: good corporate citizens

– Not-so-intentional: Misaligned incentives within firms

• Static v. dynamic strategies. (Focus on present v.

future)


Production Technology

Definition: A firm’s technology specifies a quantitative

relationship between combinations of inputs and the

level of output.

Important aspects of technology:

– Marginal product

– Factor substitution

– Economies of scale

– Short and long run

We will see a few ways to describe technology (function,

graph).



Examples of production technology:

• The recipe for Coca-Cola

• GE Appliances

Given 10 hours of labor, a factory, a specific collection of machinery

and parts, GE can produce a certain number of washing machines.

• Scrambled eggs

It takes 2 eggs, ¼ cup of milk, ¼ tsp each of salt and pepper, a wisk,

a pan, heat, and 10 minutes of labor to make 1 serving of scrambled

eggs.

The last example should make it clear that it is hard to describe

real-world production fully without a ton of detail. Simplify!



To deal with this, we consider a single output and two inputs:

Labor (L):

• Stands in for all inputs that can be varied immediately. (We

will sometimes just say “variable inputs.”)

Capital (K):

• Represents all inputs that might not be able to change

immediately. (We will sometimes just say “fixed inputs.”)

Important: What is variable and fixed changes over time.


Production Functions

General form: Q = f(L,K,…)

• What f(L,K) tells us: Amount of Q that comes out when L and

K go in.

• An assumption: f(L,K) tells us the maximum quantity that

comes out when L and K go in.

• Production functions can be easily tailored to include other

types of inputs or a greater number of them.

Examples

1. A utility produces electricity from coal and/or natural gas. Each ton of coal (C) produces 60 kilowatt hours, each 100 cubic meters of gas (G) produces 40 kilowatt hours.

Production function: Q = 60C + 40G



Examples

2. The number of insurance claims processed in a day is

Q = K1/2L1/2

K = total computer power, L = number of claims workers.

If K = 2, L = 2: Q = 2

K = 9, L = 1: Q = 3

K = 9, L = 4: Q = 6



Examples

3. John and Paul create songs. To create 4 minutes of music,

John (J) needs to work for 2 hours and Paul (P) works for 3

hours. Without both John and Paul working on a song,

nothing of value is created.

The production function is Q = 4min{J/2, P/3}

How much music is produced when J = 15 and P = 12?


Production FunctionsWe have now seen production functions in which…

• Inputs are perfectly substitutable: Q = 60C + 40G

• Inputs are perfectly complementary: Q = 4min{J/2, P/3}

• Inputs are imperfectly substitutable: Q = K1/2L1/2

Additional facts about Cobb-Douglas technology, Q = kLaKb:

• The relative size of a and b affect the relative productivity of

L and K

• The sum (a+b) determines economies of scale

• The term k cannot be ignored (as in utility), as it shifts Q in a

tangible way.


Illustrating Production Tech.Just as we used indifference curves to illustrate preferences, we

can draw isoquants to show how different combinations of

inputs can yield the same output quantity.

K

Q = 60

L

Q = 90

Q = 40

2 3

2

3

Assume a firm has Q = 10LK.

How can it combine L and K to

produce a variety of Q levels?


Illustrating Production Tech.Example:

• Return to the electricity production function: Q = 60C + 40G

• Draw the isoquant for 1200 kilowatt hours of the electricity

production technology.

G

C20

30• Where do the intercepts

come from?

• What does the slope of

the isoquant mean?


Illustrating Production Tech.Example:

• Return to the music production function: Q = 4min{J/2,P/3}

• Draw the isoquant for 40 minutes of music production.

P

J

• How do we interpret the

kink?

• How will J and P choose

to allocate their time?

20

30


Marginal & Average Product

Given fixed K, what’s the

benefit to adding L?

L K Q Q/L dQ/dL

0 10 0 - -

1 10 10 10 10

2 10 30 15 20

3 10 60 20 30

4 10 80 20 20

5 10 95 19 15

6 10 108 18 13

7 10 112 16 4

Average

Product

of Labor

Marginal

Product

of Labor


Marginal Product

Marginal Product of Labor (MPL)

Increase in output from one more L, holding K fixed.

Marginal Product of Capital (MPK)

Increase in output from one more K, holding L fixed.

• We generally assume MPL & MPK > 0.

• MPs are important for choosing the optimal L and K.


Marginal Product

Example: Q = L0.5K0.5

Suppose L = 9 and K = 4, so Q = 6.

What’s the benefit from adding another worker or unit of

capital?

MPL = 0.5L-0.5K0.5 = 1/3.

Interpretation: With another unit of L, Q increases to 6.33.

MPK = 0.5L0.5K-0.5 = 0.75

Interpretation: With another unit of K, Q increases to 6.75.


Marginal Product

Example: Q = min{3L, 2K}

Suppose L = 2 and K = 2, so Q = 4.

What’s the benefit from adding another worker or unit of

capital?

MPL = 0

Interpretation: Without additional K, more L does no good.

MPK = 2

Interpretation: With another unit of K, Q increases to 6.


Marginal Product

Law of Diminishing Marginal Returns

• Eventually, the marginal product of every input declines.

• This means that MPL (or MPK) eventually falls. It does

NOT mean that it must become negative.

• Also, it is NOT the case that later L are lower quality

(e.g., less educated) than initial L.

Example: Q = L0.5K0.5

• What is MPL at K = 4? MPL = L-0.5

• How does this change as L ? Bigger L Smaller MPL


Marginal and Average Product

Average Product of Labor

• This is “Labor Productivity.” International comparisons of

productivity use this measure. (This is a short run concept.)

• What are the main determinants of this?

• What determines productivity growth in services, agriculture?

Marginal, Average Products of Capital

• Note that we can define the same concepts for capital (K)

• This could be because we want to treat K as flexible relative

to L, plus we need the concepts for long run analysis.


Illustrating Productivity

Q

L

MPL

APL

L

MPL

APL

For any production function, we can illustrate MPL and/or

APL, given a level of K

( , )f L K

Notice that APL increases when MPL is above it, and APL

falls when MPL is below it.



Another look at the relationship between MPL and APL:

Q

L

Q0

L0

( , )f L K

• APL at L0 is Q0/L0 = Slope

of red dashed line.

• How do we illustrate MPL

for this production fn?

• Interpretation of why APL

and MPL are different

here?



For higher levels of K, we will usually see higher levels of MPL.

MPL

L

MPL1

MPL2

• MPL1 is associated

with a low level of K

• MPL2 is associated

with higher K

• What do we make of

the way MPL1 and

MPL2 differ (more at

high L than low L)?


Factor Substitution

K

Q = 10

L

Slope of an isoquant:

Marginal Rate of Technical

Substitution (MRTS)

MPL

MRTSMPK

Interpretation: Holding Q fixed, how much K is the firm

willing to give up for one more unit of L?

Various L and K can lead to the same Q when inputs are

substitutable. This follows from MPL > 0 and MPK > 0.


Factors Substitution

Why are we sure that MRTS is the slope of the isoquant?

• MPx is the magnitude of change in output if input x goes up

or down by a small amount.

• If L increases by dL, then Q increases by: MPL dL

• If K decreases by dK, then Q falls by: – MPK dK

In general,

• MPLdL + MPKdK = dQ. This holds for (+) or (–) dL & dK.

• Since we are moving along an isoquant, we are interested in

dQ = 0. Insert this and rearrange to get:

MPL dK

MPK dL


Variation in Factor Substitution• When K is high and L is low, MRTS may be high. Why?

Look at MPL and MPK.

• Diminishing MRTS: Flexibility in exchanging K for L becomes

more difficult as the initial amount of K falls.

K

Q = 10

L

1

2

MRTS = MPL/MPK

MRTS1 > MRTS2

because of differences

in labor and capital

productivity.


Scale Economies

A firm’s efficiency may depend on how large it is. What do

we mean by efficiency?

High output for relatively low inputs

Example: 3 Restaurants – Which is most efficient?

a. Small. One chef in a small kitchen.

– Must do everything himself & use limited equipment

b. Medium. Ten chefs in a kitchen 10 as large as in (a).

– Tasks divided among chefs saves time for all.

Specialized capital.

– More than 10 the output of (a)


Scale Economies

Restaurant example, continued

c. Large. 100 chefs in a huge kitchen, 10 size of (b).

– Fine division of tasks requires a lot of coordination. No

additional benefits from specialized capital.

– Less than 10 the output of (b).

Would you say the small, medium, or large restaurant is most

efficient?


Quantifying Scale Economies

Suppose we have the production function f(L,K).

Steps in considering the scale economies of f :

1. Start with certain amounts of the inputs, L1 and K1.

2. Now think about scaling-up the inputs by a factor >

1, so that L2 = L1 > L1 and K2 = K1 > K1

Example: = 1.5, to represent a 50% increase in L and K.

3. What happens to output? (Does it increase by more

or less than ?)



How to do #3: Compare Q2 = f(L1, K1) to Q1 = f(L1,K1).

The function f has…

• Increasing returns to scale if Q2/Q1 > .

• Constant returns to scale if Q2/Q1 = .

• Decreasing returns to scale if Q2/Q1 < .

Examples: What are the returns to scale for…

• Q = 3L + 2K

• Q = 3L1/2 + 2K1/2



• Example: Cobb-Douglas production Q = LaKb.

• L and K increase by . What happens to output?

• Compare to Q1 = LaKb. What determines Q2/Q1 ?

• Whether a+b > depends on how (a+b) compares to 1.

E.g.: if (a+b) > 1, then f has increasing returns to scale.

2 ( ) ( )

( )( )

a b

a b a b

Q L K

L K

2

1

( )( )

( )

a b a ba b

a b

Q L K

Q L K


Illustrating Scale EconomiesWe measure scale economies by asking how much output

increases if inputs increase by some factor (e.g., double).

If inputs double…

• Double the output:

Constant returns to scale

• More than double:

Increasing returns to scale

• Less than double:

Decreasing returns to scale

K

Q = 10

L

Q = ??

8 16

16

8


Scale Economies

• Most real-world technologies are believed to have…

– increasing returns to scale (RTS) for low Q, and

– decreasing RTS for high Q.

• Why shouldn’t we expect to observe many increasing

RTS situations at firms’ observed sizes?

General intuition:

• In long-run equilibrium, we should expect firms to

operate at their most efficient scale.

• It’s no accident that coffee shops are small and auto

plants are huge.


Scale Economies

Example: Decreasing RTS in microprocessor production.

• Microprocessors produced in clean “fabs”

• Fabs are often the same size

• Increasing returns for very small fabs, but as fabs get

larger is it difficult to keep them clean and organized.

• Intel has many fabs.

More generally, the average U.S. manufacturing firm has 4

similarly sized plants.


Short Run and Long Run

Suppose that a firm is “locked in” to a fixed level of capital

for the immediate future. Examples:

- Changing capital levels would mean building a new factory

or selling-off an old one, and this doesn’t happen quickly.

- New high-tech machinery can be obtained immediately,

but it takes a while for workers to learn how to use it.

What this means for the firm:

- The firm should try to use its current capital as well as

possible.

- The firm must pay for all of the capital it currently has.


Short Run and Long Run

Given a fixed level of capital, K1, in the short run, how

difficult is it to change output goals?

K

Q = 10

L

Q = 20

K1

L1

• Suppose a production goal

changes from Q = 10 to 20.

• Given the blue isoquants,

what new level of L is needed

to achieve the new goal?

• Given the red isoquants, what

is the new level of L?

• Which technology (set of

isoquants) allows more

substitution between inputs?LB LR


Summary

We have covered:

• How to describe a production technology

• Important details:

– Marginal and average productivity

– Scale economies

– Long run v. short run flexibility

This will be useful to us because:

• Firms costs follow directly from productivity.

• Changes to firms’ organizational strategies depend on

what’s possible under the production technology.

Production

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Transcript of Production