production
TRANSCRIPT
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.
Production Technology 2
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…)
Production Technology 3
What do firms choose and why?
Constraints:
• Technology
• Budget (-like) constraints from input and output markets.
Production Technology 4
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?
Production Technology 5
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.
Production Technology 6
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
Production Technology 7
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 8
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).
Production Technology 9
Production Technology
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!
Production Technology 10
Production Technology
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 Technology 11
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
Production Technology 12
Production Functions
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
Production Technology 13
Production Functions
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 Technology 14
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.
Production Technology 15
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?
Production Technology 16
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?
Production Technology 17
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
Production Technology 18
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
Production Technology 19
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.
Production Technology 20
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.
Production Technology 21
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.
Production Technology 22
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
Production Technology 23
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.
Production Technology 24
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.
Production Technology 25
Illustrating Productivity
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?
Production Technology 26
Illustrating Productivity
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)?
Production Technology 27
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.
Production Technology 28
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
Production Technology 29
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.
Production Technology 30
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)
Production Technology 31
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?
Production Technology 32
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 ?)
Production Technology 33
Quantifying Scale Economies
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
Production Technology 34
Quantifying Scale Economies
• 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
Production Technology 35
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
Production Technology 36
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.
Production Technology 37
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.
Production Technology 38
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.
Production Technology 39
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
Production Technology 40
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.