Theory of Production

and Cost Analysis

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Production
Theory

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Learning Objectives
Production Functions
▪ Use a production function to describe the relationship between inputs
and output
Short-Run Production
▪ Predict the effects of short-run changes in labor on output
Long-Run Production
▪ Explain the long-run trade-off between labor and capital in production
Returns to Scale
▪ Determine whether a production function has decreasing, constant, or
increasing returns to scale

Innovation
▪ Describe the effects of innovation on production

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Managerial Problem
Labor Productivity During Recessions

• How much will the output produced per worker rise or fall with each additional layoff?

Solution Approach

• First, a firm must decide how to produce. Second, if a firm wants to expand its output, it
must decide how to do that in the short and the long run. Third, given its ability to
change its output level, a firm must determine how large to grow.

Empirical Methods

• A production function summarizes how a firm converts inputs into outputs using one
available technology. It helps to decide how to produce.

• Increasing output in the short run can be done only by increasing variable inputs. In the
long run, there is more flexibility.

• The size of a firm depends on returns to scale. Its growth is determined by increments
in productivity that comes from technological change.

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Production Functions (1 of 2)
Production Process
• A firm uses a technology or production process to transform inputs or
factors of production into outputs.
• Inputs
– Capital (K)—land, buildings, equipment
– Labor (L)—skilled and less-skilled workers
– Materials (M)—natural resources, raw materials, and processed products
• Output
– It could be a service, such as an automobile tune-up by a mechanic, or a
physical product, such as a computer chip or a potato chip.
• Production function

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Production Functions (2 of 2)
• The production function summarizes ways inputs = output

– Maximum quantity of output that can be produced with different combinations of

inputs, given current knowledge about technology and organization.
– A production function shows only efficient production processes because it
gives the maximum output.

• Production Function: q = f (L, K )

– Production function for a firm that uses only labor and capital.
– q units of output (wrapped candy bars) are produced using L units of labor services
(hours of work by assembly-line workers) and K units of capital (number of
conveyor belts).

• Production Functions, Time and Variability of Inputs

– Short run: a period of time where at least one factor of production cannot be
varied. Inputs can be fixed or variable.
– Long run: period of time that all relevant inputs can be varied. Inputs are all
variables.

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Short-Run Production (1 of 5)
• The short-run is a period in which at least one input is fixed. In the context of a
production process with only capital and labor as inputs, we assume that capital is fixed
and labor is variable.
The Total Production Function: q = f (L, K )

• Relationship between output q and labor L when a firm’s capital K is fixed

– In Table 5.1, capital is fixed to 8 fully equipped workbenches. As the number of
workers increases, so does output: 1 worker assembles 5 computers in a day, 2
workers assemble 18, 3 workers assemble 36, and so forth.
The Average Product of Labor: APL = q / L

• The ratio of output to the amount of labor used to produce that output
– Table 5.1 shows that 9 workers assemble 108 computers a day, so the average
product of labor for 9 workers is 12 ( = 108 / 9 ) computers.

– Ten workers can assemble 110 computers in a day, so the average product of
labor for 10 workers is 11 ( = 110 / 10 ) computers.

• Thus, increasing the labor force from 9 to 10 workers lowers the average product per
worker.
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Table 5.1
Total Product, Marginal Product, and Average Product of Labor with Fixed Capital

Average Product of Labor,

Output, Total Marginal Product of Labor,
Capital, K Labor, L Average Product of
M P sub L = delta q

Product of Labor q MPL = q / L Labor, APL = q / L

K bar

over delta L

A P sub L = q over L

Blank Blank

8 0 0
8 1 5 5 5
8 2 18 13 9
8 3 36 18 12
8 4 56 20 14
8 5 75 19 15
8 6 90 15 15
8 7 98 8 14
8 8 104 6 13
8 9 108 4 12
8 10 110 2 11
8 11 110 0 10
8 12 108 −2 9
8 13 104 −4 8

Labor is measured in workers per day. Capital is fixed at eight fully equipped workbenches.

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Short-Run Production (2 of 5)
The Marginal Product of Labor: MPL = q / L

• Change in total output resulting from using an extra unit of labor, holding other factors
(capital) constant.

– Table 5.1 shows if the number of workers increases from 1 to 2, L = 1,

output rises by q = 13 = 18 − 5, so the marginal product of labor is 13.

• When the change in labor is very small (infinitesimal), we use the calculus definition of
the marginal product of labor.
– Marginal Product of Labor is the partial derivative of the production function with
respect to labor:

– MPL = q / L = f (L, K ) / L

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Short-Run Production (3 of 5)
Graphing the Product Curves

• Figure 5.1 plots the data from Table 5.1. The figure shows how output (total product),
the average product of labor, and the marginal product of labor vary with the number of
workers.

The Effect of Extra Labor

• In panel a of Figure 5.1, output rises with labor until it reaches its maximum of 110
computers at 11 workers, point C.

• In panel b, the average product of labor first rises and then falls as labor increases.
Also, the marginal product of labor first rises and then falls as labor increases.

• Average product may rise because of division of labor and specialization. Workers
become more productive as we add more workers. Marginal product of labor goes up,
and consequently average product goes up.

• Average product falls as the number of workers exceeds 6. Workers might have to wait
to use equipment or get in each other’s way because capital is constant. Because
marginal product of labor goes down, average product goes down too.

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Figure 5.1
Production Relationships with Variable Labor

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Short-Run Production (4 of 5)
Relationships Among Product Curves
• The total, average and marginal curves are geometrically related.
• Average Product of Labor and Marginal Product of Labor
– If the marginal product curve is above that average product curve, the average
product must rise with extra labor.
– If marginal product is below the average product then the average product must fall
with extra labor.
– Consequently, the average product curve reaches its peak where the marginal
product and average product are equal (point a in panel b of Figure 5.1).
• Deriving AP and MP
L L using the Total Production Function
– The average product of labor for L workers equals the slope of a straight line from
the origin to a point on the total product of labor curve for L workers in panel a of
Figure 5.1.
– The slope of the total product curve at a given point equals the marginal product of
labor. That is, the marginal product of labor equals the slope of a straight line that
is tangent to the total output curve at a given point.

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Short-Run Production (5 of 5)
The Law of Diminishing Marginal Returns

• If a firm keeps increasing an input, holding all other inputs and technology
constant, the corresponding increases in output will eventually become smaller
(diminish).
– This law comes from realizing most observed production functions have
this property.
– Common Confusion: “An input’s marginal product must eventually fall as
a firm uses more of the input” is true only in the short run.

• The law of diminishing marginal returns determines the shape of the marginal
product of labor curves:
– if only one input is increased, the marginal product of that input will
diminish eventually.

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Long-Run Production (1 of 4)
• In the long run, labor L and capital K are variable.
• The firm can substitute one input for another while continuing to produce the same
level of output, q .

Isoquants: q = f (L, K )

• An isoquant shows the efficient combinations of labor and capital that can produce the
same (iso) level of output (quantity).
– In table 5.2, output q is produced with the combination of labor L and capital K.
– In Figure 5.2, points a, b, c, and d show the same level of output, q = 24. This level
of output is also highlighted in Table 5.2

Properties of Isoquants
• The farther an isoquant is from the origin, the greater is the level of output.
• Isoquants do not cross.
• Isoquants slope downward.
– All these properties can be verified in Figure 5.2.

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Table 5.2
Output Produced with Two Variable Inputs

Blank Labor, L Labor, L

Labor, L Labor, L Labor, L Labor, L

Capital, K 1 2 3 4 5 6
1 10 14 17 20 22 24
2 14 20 24 28 32 35
3 17 24 30 35 39 42
4 20 28 35 40 45 49
5 22 32 39 45 50 55
6 24 35 42 49 55 60

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Figure 5.2
A Family of Isoquants

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Long-Run Production (2 of 4)
Shapes of Isoquants

• If inputs are perfect substitutes, isoquants are straight lines.

– In Figure 5.3, panel a, potato salad can be produced with potatoes from
Idaho or Maine, both are perfect substitutes. Isoquants are straight lines
with a slope of −1.

• If inputs cannot be substituted at all, so inputs must be used in fixed

proportions, isoquants are right angles.
– In Figure 5.3, panel b, the production of 12-ounce boxes of cereal
requires 1 box and 12-ounce of cereal, and this proportion is fixed. The
efficient points of production are in the 45˚line and the dashed lines are
inefficient points.

• If inputs can be substituted imperfectly, isoquants are convex to the origin.

– In Figure 5.3, panel c, the isoquant at level q=1 is convex, and the slope
varies at each point of the isoquant.

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Figure 5.3
Substitutability of Inputs

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Long-Run Production (3 of 4)
Substituting Inputs: MRTS = K / L

• The slope of an isoquant shows the ability of a firm to replace one input with
another while holding output constant.
• This slope is the marginal rate of technical substitution (MRTS): how many
units of capital the firm can replace with an extra unit of labor while holding
output constant.
Substitutability of Inputs Varies Along and Isoquant
• Diminishing MRTS (absolute value): The more labor and less capital the firm
has, the harder it is to replace remaining capital with labor and the flatter the
isoquant becomes.
– In Figure 5.4, the firm replaces 6 units of capital per 1 worker to remain on
the same isoquant (a to b), so MRTS = −6. If the firm hires another
worker (b to c), the firm replaces 3 units of capital, so MRTS = −3.

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Figure 5.4
How the Marginal Rate of Technical Substitution Varies
Along an Isoquant

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Long-Run Production (4 of 4)
Substitutability of Inputs and Marginal Products

• The marginal rate of technical substitution is equal to the ratio of marginal

products

– −MPL / MPK = K / L = MRTS

 
Cobb-Douglas Production Functions: q = AL K

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Returns to Scale
• How much output changes if a firm increases all its inputs proportionately?
There are three possible answers:
Constant Returns to Scale (C R S): f ( 2L, 2K ) = 2f ( L, K ) = 2q

• A technology exhibits constant returns to scale if doubling inputs exactly

doubles the output. The firm builds an identical second plant and uses the
same amount of labor and equipment as in the first plant.

Increasing Returns to Scale (I R S): f ( 2L, 2K )  2f ( L, K ) = 2q

• A technology exhibits increasing returns to scale if doubling inputs more than

doubles the output. Instead of building two small plants, the firm decides to
build a single larger plant with greater specialization of labor and capital.
Decreasing Returns to Scale (D R S): f ( 2L, 2K )  2f ( L, K ) = 2q

• A technology exhibits decreasing returns to scale if doubling inputs less than

doubles output. An owner may be able to manage one plant well but may have
trouble organizing, coordinating, and integrating activities in two plants.

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Varying Scale Economies

https://neb11help.wordpress.com/2021/04/03/economics-notes-theory-
of-production/

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Varying Scale Economies

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Returns to Scale
Managerial Implication: Small Is Beautiful
• Over the years, the typical factory has
grown in size to take advantage of
increasing returns to scale.
• However, 3 D printing may reverse this trend
by making input requirements per unit to
manufacture one item as low as when
making thousands.

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 https://akfpartners.com/growth-
blog/category/leadership-and-management

https://akfpartners.com/growth-
blog/category/leadership-and-management
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Returns to Scale
Managerial Implication: Small Is Beautiful

• Over the years, the typical factory has grown in size to take advantage of
increasing returns to scale. However, 3D printing may reverse this trend by
making input requirements per unit to manufacture one item as low as when
making thousands.

• 3D printing is now used in manufacturing final products, particularly in

industries that need small numbers of customized parts.
– Nike is using it to produce the Zoom Superfly Skyknit running shoe.
Biomedical and aerospace companies like Boeing and Airbus use it for
just-in-time manufacturing. Airbus introduced the world’s first entirely 3D-
printed aircraft, a drone, in 2016.

• Managers should experiment with 3D. They can produce small initial runs to
determine the size of the market and consumers’ acceptance of the product.
Based on information from early adopters, managers can determine if the
market warrants further production and can quickly modify designs to meet
end-users’ desires.

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Innovation (1 of 2)
• Innovation, a new idea, device or method, affects markets in many ways.

• We, consumers, are familiar with product innovations. But in the context of production,
we focus on process and organizational innovations.

Process Innovation (new method of production)

• A process innovation changes the production function: more output to be produced

with the same level of inputs (technological progress).
– Technological progress is neutral if more output is produced using the same ratio
of inputs.
– Technological progress is nonneutral if it is capital or labor saving.

Organizational Innovation (new way of organizing)

• Organizational innovations may also alter the production function and increase the
amount of output produced by a given amount of inputs.
– In the early 1900s, Henry Ford revolutionized mass car production through
interchangeable parts and the assembly line.

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Innovation (2 of 2)
Managerial Implication:
Technical Progress and Competitive Advantage
• In some industries, an important part of being a good
manager involves adopting new technologies and looking
for better ways to organize the firm.
• A firm that achieves a successful process or organizational
innovation can produce more output with the same input
levels than its rivals, which gives it a competitive
advantage.

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Managerial Problem Solution
Labor Productivity During Recessions
• How much will the output produced per worker rise or fall with each additional
layoff?

Solution
• Layoffs have the positive effect of freeing up machines to be used by
remaining workers. However, if layoffs force the remaining workers to perform
a wide variety of tasks, the firm will lose the benefits from specialization.
• Holding capital constant, a change in the number of workers affects a firm’s
average product of labor. Labor productivity could rise or fall.
• For some production functions layoffs always raise labor productivity because
the AP curve is everywhere downward sloping, for instance the
L

Cobb-Douglass production function.

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Cost Theory

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Learning Objectives
The Nature of Costs
▪ Explain why managers should use opportunity costs in decision
making
Short-Run Costs
▪ Draw marginal cost, average cost, and other cost curves
Long-Run Costs
▪ Explain how to choose inputs to minimize cost
The Learning Curve
Predict how experience-based learning reduces costs
The Costs of Producing Multiple Goods
Describe when it pays to produce two or more goods simultaneously

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 Managerial Problem

Technology Choice at Home Versus Abroad

• In the United States, firms use relatively capital-intensive technology.
• Will that same technology be cost minimizing if firms move their production
abroad?

Solution Approach
• First, a firm must determine which production processes are technically
efficient so that production has no waste. Second, a firm should pick from
these technically efficient processes the one that is also economically
efficient. By minimizing costs, a firm can increase its profit.

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Managerial Problem

Empirical Methods
• When considering costs, a good manager includes
opportunity costs.
• To minimize costs, a manager should distinguish short-
run from long-run costs.
• Firms may reduce costs overtime based on experience or
its learning curve.
• If a firm produces several goods, individual cost may
depend on the cost of producing multiple goods.

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The Nature of Costs
• Financial accounting statements correctly measure costs
for tax purposes and to meet other legal requirements.
• To make sound managerial decisions, good managers
need more information and a perspective about explicit
and implicit costs.
– Explicit costs are direct, out-of-pocket payments for
labor, capital, energy, and materials.
– Implicit costs reflect only a foregone opportunity
rather than explicit, current expenditure.

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The Nature of Costs…2
Opportunity Costs
• The opportunity cost of a resource is the value of the best alternative
use of that resource.

– Maoyong owns and manages a firm. He pays himself only $1k per
month but could work for another firm and make $11k per month.
Working for another firm is the best alternative use of his time, so
his opportunity cost of time is $11k.

– Assume monthly revenue is $49k and explicit costs are $40k,

including Maoyong’s monthly wage. The accounting profit is $9k
and Maoyong collects $10k per month (profit + wage). However,
his opportunity cost is $11k. So, he incurs an economic loss of
$1k.

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The Nature of Costs…..3
Bottomline
• There’s no such thing as a free lunch” captures the
concept of opportunity cost.
– Even if you are invited for free lunch by your parents,
you still incur the opportunity cost of your time.
• Common Confusion: I can save money by doing things
myself rather than buying goods and services from firms.
– The fallacy in this belief is that we have ignored the
opportunity cost of our time.

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The Nature of Costs….4
Costs of Durable Inputs

• Durable inputs are usable for a long period, perhaps for many years.
– Capital such as land, buildings, or equipment are durable inputs.

• Two problems with the costs of durable inputs

– How to allocate the initial purchase cost over time?
– What to do if the value of the capital changes over time?

• Solutions to the calculation of durable inputs (truck example)

– If there is a rental market: The accountant may expense the truck’s
purchase price or may amortize it over the life of the truck, following I R S
rules. The firm’s opportunity cost of using the truck is the amount that
the firm would earn if it rented the truck to others.
– If there is no rental market: The opportunity cost of capital of using the
truck a year would be the interest forgone in a year.

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The Nature of Costs…5
Sunk Costs
• Sunk cost is a past expenditure that cannot be recovered.
– If an expenditure is sunk, it is not an opportunity cost. So we
should not consider it for managerial decisions.
– However, sunk costs appear in financial accounts.
• Consider a firm buys a forklift for $30k,
– If the firm can resell it for $30k, then the expenditure is not sunk,
and the opportunity cost of using the forklift is $30k.
– If the firm cannot resell it, then the original expenditure is a sunk
cost—it cannot be recovered, and its opportunity cost is zero. The
forklift should not be included in the firm’s current cost
calculations.
– If the firm can resell it for $20k, then only $10k of the original
expenditure is sunk, and the opportunity cost is $20k.

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 Managerial Implication:
Ignoring Sunk Costs

• A manager should ignore

sunk costs when making
current decisions.

• In making its decisions, use

the opportunity cost and
ignore the sunk cost.

• Don’t cry over spilt milk.

What’s done is done. Don’t
throw good money after
bad.

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Managerial Implication: Ignoring Sunk
Costs
• A manager should ignore sunk costs
when making current decisions.
• In making decisions, use the opportunity
cost and ignore the sunk cost.
• Don’t cry over spilt milk. What’s done is
done. Don’t throw good money after bad.

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Short-Run Costs
Common Measures of Cost: Fixed, Variable, and Total

• All firms use the same basic cost measures, and these should be
based on input’s opportunity costs.
• Fixed cost (F) does not vary with the level of output; includes
expenditures on land, office space, production facilities, and other
overhead expenses; are often sunk costs, but not always.
• Variable cost (VC) changes as the quantity of output changes; refers
to the costs of variable inputs.
• Total cost (C) is the sum of fixed and variable costs.
• F and VC should be based on inputs’ opportunity costs.
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Short-Run Costs………..2
Average Cost

• Average fixed cost (A F C) falls as output rises because the fixed cost is spread over
more units.
• Average variable cost (A V C) or variable cost per unit of output may either increase or
decrease as output rises.
• Average cost (A C) or average total cost may either increase or decrease as output
rises.
– In Table 6.1, A FC falls with output and A V C eventually rises with output. A C falls
until output of 8 units and then rises.

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Short-Run Costs….3
Marginal Cost
• MC = C / q
– Marginal cost (M C) is the amount by which a firm’s cost changes if the firm
produces one more unit of output.
– ΔC is the change in cost when the change in output, Δq, is 1 unit.
• MC = VC / q

– Marginal cost also equals the change in variable cost from a one-unit increase in
output.
– VC / q is the change in variable cost when the change in output, Δq, is 1 unit.
– In Table 6.1, if the firm increases output from 2 to 3 units, the marginal cost is $20.

• Marginal Cost using Calculus: MC = dC / dq = dVC / dq

– Marginal cost is the rate of change of cost as we make an infinitesimally small
change in output.
– MC = dVC / dq because dF / q = 0.

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Table 6.1
How Cost Varies with Output
Output, q Fixed Variable Total Marginal Average Fixed Average Average Cost,
Cost, F Cost, V C Cost, C Cost, M C Cost, Variable Cost,
AC = C /q
A C = C over q

AFC = F /q
A F C = F over q A V C = V C over q

AVC = VC /q
0 48 0 48 blank blank blank blank

1 48 25 73 25 48 25 73
2 48 46 94 21 24 23 47
3 48 66 114 20 16 22 38
4 48 82 130 16 12 20.5 32.5
5 48 100 148 18 9.6 20 29.6
6 48 120 168 20 8 20 28
7 48 141 189 21 6.9 20.1 27
8 48 168 216 27 6 21 27
9 48 198 246 30 5.3 22 27.3
10 48 230 278 32 4.8 23 27.8
11 48 272 320 42 4.4 24.7 29.1
12 48 321 369 49 4 26.8 30.8

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Short-Run Costs….4
Cost Curves

• Based on Table 6.1, panel a of Figure 6.1 shows the V C, F, and C curves.
– The fixed cost curve, F, is a horizontal line at $48.
– The variable cost curve, V C, is zero at zero units of output and rises with output.
– The total cost curve, C, is the vertical sum of the V C and F curves, so it is $48
higher than the V C curve at every output level. V C and C curves are parallel.

• Panel b in Figure 6.1 shows the A FC, A V C, A C, and M C curves.

– The marginal cost curve, M C, cuts the average variable cost, A VC, and
average cost, AC, curves at their minimums.
– The height of the AC curve at point a equals the slope of the line from the origin to
the cost curve at A.
– The height of the AV C at b equals the slope of the line from the origin to the V C
curve at B.
– The height of the M C is the slope of either the C or V C curve at that quantity.

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Short-Run Costs………5
Production Functions and the Shapes of Cost Curves
• The production function determines the shape of a firm’s cost curves.
• In the short run, diminishing marginal returns to labor determine
the shape of the production function.
– The firm increases output by using more labor. However, each
extra worker increases output by a smaller amount.
• The production function determines the shape of the variable
cost curve and its related curves.
– As output increases, variable cost increases more than
proportionally because of diminishing marginal returns.
– The production function determines the shape of the marginal
cost, average variable cost, and average cost curves.

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 Short-Run Costs………..6
The Variable Cost Curve

• If input prices are constant, the firm’s

production function determines the shape
of the variable cost curve.
– The VC and the total product curve
have the same shape, as it is shown
in Figure 6.2.
– The firm faces a constant input price
for labor, $10 per hour.
– The total product curve uses the
horizontal axis measuring hours of
work.
– The variable cost curve uses the
horizontal axis measuring labor cost:
VC = wL.

– The VC of 6 units of output is $240 ( $10 * 24 ) .

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Figure 6.2
Variable Cost and Total Product

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Short-Run Costs………7
The Marginal Cost Curve
• MC = VC / q
– Marginal cost (M C) is the change in V C as output increases by 1 unit.
– The M C curve is U-shaped because of diminishing marginal returns.
• MC = VC / q = w ( L / q )
– In the short run, capital is fixed. So, the change in V C as output increases by 1 unit
must be the change in the cost of labor.
– Marginal cost equals the wage times the extra labor necessary to produce 1
more unit of output.
– In Figure 6.2, to increase Q from 5 to 6 units, MC = $40(4  $10).

• MC = w / MPL
– Remember MPL = q / L. So, L / q is just the inverse of MPL .
– Marginal cost equals wage divided by marginal product of labor.
– Marginal product of labor and marginal cost move in opposite directions as output
changes.

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Short-Run Costs…..8
The Average Cost Curves
• AVC = VC / q

– Average variable cost is just the variable cost divided by output.

– Given the shape of the V C is determined by the production function, the
diminishing marginal returns to labor also determines the shape of the A V C cost
curve.
– So, the V C and A VC curves are both U-shaped.
• AVC = VC / q = wL / q
– In the short run, capital is fixed. So the variable cost is wL.
– The average variable cost is cost of labor divided by output.
• AVC = w / APL
– Remember APL = q / L. So, L / q is just the inverse of APL .
– Average cost equals wage divided by average product of labor.
– In Figure 6.2, at 6 units of Q, the AVC = $10 / 0.25 = $40.
– Average product of labor and average variable cost move in opposite directions as
output changes.

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Short-Run Costs……9
Short-Run Cost Summary
1. In the short run, the cost associated with fixed inputs is fixed,
while the cost from inputs that can be adjusted is variable.
2. Given that input prices are constant, the shapes of the variable
cost and the cost-per-unit curves are determined by the
production function.
3. Where there are diminishing marginal returns, the variable cost
and cost curves become relatively steep as output increases,
so the average cost, average variable cost, and marginal cost
curves rise with output.
4. The average cost and average variable cost curves fall when
marginal cost is below them and rise when marginal cost is
above them, so the marginal cost cuts both these average cost
curves at their minimum points.
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Long-Run Costs
• In the long run, the firm adjusts all its inputs so that its cost of production is as
low as possible.
– The firm can change its plant size, design, build new machines, and
otherwise adjust inputs that were fixed in the short run.
– Fixed costs are avoidable in the long run. They are not sunk costs, as
they are in the short run. For instance, the rent a restaurant pays is a
fixed cost and this rent can be avoided in the long run if the restaurant
does not renew the rental agreement.

Input Choice

• Technically and Economically Efficient

– From among the technically efficient combinations of inputs that can be
used to produce a given level of output, a firm wants to choose that
bundle of inputs with the lowest cost of production, which is the
economically efficient combination of inputs.
– To do so, the firm combines information about technology from the
isoquant with information about the cost of production.
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Long-Run Costs
The Isocost Line: C = wL + rK
• An isocost represents all the combinations of inputs that have the same total cost, C.
– Five of the many combinations of labor and capital that the firm can buy for $200
are in Table 6.2.
– In Figure 6.3, the $200 isocost line represents all the combinations of labor and
capital that the firm can buy for $200.

• Properties of Isocosts
– The points at which the isocost lines hit the capital and labor axes depends on the
firm’s cost and on the input prices
– Isocost lines that are farther from the origin have higher costs than those closer to
the origin.
– The slope of each isocost line is the same: K / L = −w / r , the rate at which
the firm can trade capital for labor in input markets.
– All these properties can be verified by looking the isocots in Figure 6.3.

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Table 6.2
Bundles of Labor and Capital That Cost the Firm $400

Bundle Labor, L Capital, K Labor Cost, Capital Cost, Total Cost,

wL = $20L rK = $40K wL = rK
a 20 0 $400 $0 $400

b 14 3 $280 $120 $400

c 10 5 $200 $200 $400

d 6 7 $120 $280 $400

e 0 10 $0 $400 $400

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Figure 6.3
A Family of Isocost Lines

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Long-Run Costs
Combining Cost and Production Information

• The firm minimizes its cost by using the combination of inputs on the isoquant that is
on the lowest isocost line that touches the isoquant.

• Isocost and Isoquant Combined: Graph Analysis

– In Figure 6.4, the lowest possible isoquant that will allow the beer manufacturer to
produce 100 units of output is tangent to the $2,000 isocost line.
– At the bundle of inputs x, the firm uses L = 50 workers and K = 100 units of capital.

– At x, the isocost is tangent to the isoquant, so the slope of the isocost, −w / r = −3,
equals the slope of the isoquant, which is the negative of the marginal rate of
technical substitution.
– Notice, y and z also produce 100 units of output but at a cost of $3,000. The x
input combination is economically efficient.

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Figure 6.4
Cost Minimization

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Long-Run Costs
• There are three equivalent rules to minimize costs in the long run: the lowest
isocost rule, the tangency rule and the last-dollar rule.

• The Lowest Isocost Rule

– The firm minimizes its cost by using the combination of inputs on the
isoquant that is on the lowest isocost line that touches the isoquant.

• The Tangency Rule: MRTS = −w / r

– At the minimum-cost bundle, x, the isoquant is tangent to the isocost line.
The slope of the isoquant (M R T S) and the slope of the isocost are equal.

• The Last-Dollar Rule: (MPL / w ) = (MPK / r )

– Cost is minimized if inputs are chosen so that the last dollar spent on
labor adds as much extra output as the last dollar spent on capital. Thus,
spending one more dollar on labor at x gets the firm as much extra output
as spending the same amount on capital.

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Long-Run Costs
Managerial Implication: Cost Minimization by Trial and Error
• How can a manager minimize cost if the manager does not
know the firm’s production function?
• The manager can use the last-dollar rule to determine the cost-
minimizing combination of inputs through trial and error.
– The manager can experiment by adjusting each input
slightly, holding other inputs constant, to learn how
production and cost change, and then use that information
to choose a cost-minimizing bundle of inputs.
• That is, managers don’t draw isoquants and isocost lines to
make decisions. Instead, they use an insight from such an
analysis to employ the last-dollar rule.

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Long-Run Costs
Factor Price Changes

• How should the firm change its behavior if the cost of one of the factors changes?

• If one factor becomes relatively cheaper, the firm should substitute factors considering
the slopes of the isoquant and isocost curves.

• Graph Analysis:
– In Figure 6.5, the initial wage = $24 and the rental rate of capital = $8. The lowest
isocost line ($2k) is tangent to the q = 100 isoquant at x(L = 50, K = 100).
– When the wage falls from $24 to $8, the isocost lines become flatter: Labor is
relatively less expensive than capital now.
– The slope of isocost lines falls from −w / r = −24 / 8 = −3 to − 8 / 8 = −1.
– The new lowest isocost line ($1,032) is tangent at v (L = 77, K = 52).
– Thus, when the wage falls, the firm uses more labor and less capital to
produce a given level of output, and the cost of production falls from $2k to
$1,032.

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Figure 6.5
Effect of a Change in a Factor Price

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Long-Run Costs
The Shapes of Long-Run Cost Curves
• The shapes of the long-run AC and MC curves depend on the shape
of the long-run TC curve.
– In Figure 6.6, panel a, the long-run cost curve rises less rapidly
than output at levels below q* and more rapidly at higher q* levels.
As a consequence, the MC and AV costs curves are U-shaped.
Why?
– In the short-run, U-shaped forms are explained by the influence of
average fixed costs and diminishing marginal returns. Both
arguments are not valid in the long run.
• In the long run, returns to scale determine the shape of the
production function, and the production function, in turn, determines
the shape of the LRAC curve and other cost curves.

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Figure 6.6
Long-Run Cost Curves

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Long-Run Costs
• The LRAC curve must be U-shaped if a production function has
– increasing returns to scale at low levels of output,
– constant returns to scale at intermediate levels of output, and
– decreasing returns to scale at high levels of output.

• L R A C curves can have many different shapes depending whether the production
process has economies of scale or diseconomies of scale.
– A cost function shows economies of scale if the A C falls as output expands.
Increasing returns to scale in production are sufficient for it.
– If an increase in output has no effect on A C, the production process has no
economies of scale.
– A cost function shows diseconomies of scale if AC rises when output increases.
– All these relationships are illustrated in Table 6.3.

• Perfectly competitive firms typically have U-shaped L RA C curves.

• Noncompetitive markets may be U-shaped, L-shaped, everywhere downward sloping,

everywhere upward sloping or have other shapes.

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Table 6.3
Returns to Scale and Long-Run Costs

Output, Q Labour, L Capital, K Cost, C = wL + Average Cost, Returns to

rK AC = C / q
A C = C over q Scale
blank

1 1 1 24 24

3 2 2 48 16 Increasing

6 4 4 96 16 Constant

8 8 8 192 24 Decreasing

w = r = $12 per unit.

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The Learning Curve

• Average cost may fall over time because of increasing returns to scale, technological
progress or learning by doing.
• Learning by doing refers to the productive skills and knowledge that workers and
managers gain from experience.
– Workers add speed with practice. Managers learn how to organize production
more efficiently, assign tasks based on worker’s skills and reduce inventory costs.
Engineers experiment new product designs.
– For these and other reasons, the average cost of production tends to fall over time,
and the effect is particularly strong with new products.
• The learning curve is the relationship between average costs and cumulative output.
The cumulative output is the total number of units of output produced since the product
was introduced.

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Figure 6.7
Learning by Doing

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Managerial Solution
Technology Choice at Home Versus Abroad
• In the United States, firms use a relatively capital-intensive technology.
• Will that same technology be cost minimizing if firms move their production
abroad?

Solution
• The answer depends on relative factor prices and whether the firm’s isoquant
is smooth.
• If the isoquant is smooth, even a slight difference in relative factor prices will
induce the firm to shift along the isoquant and use a different technology with a
different capital-labor ratio.
• If the isoquant has kinks, the firm will use a different technology only if the
relative factor prices differ substantially.

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 Strategic
Management
Implications

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Innovation Adoption ❑ Innovation diffusion involves a
long maturation period.
Curves ❑ Accumulated adoption curve (s-
shaped)

❑ New cases curve (bell-shaped)

which is a normal distribution

❑ What do the shapes tell us about

early and late adopters?

❑ They differ from idea to idea.

❑ What separates innovators from

the others?

❑ Roger’s Five Stages:

https://www.bing.com/videos/riverv
iew/relatedvideo?q=rogers+2023+
on+innovation+adopters&mid=E22
CB5D8149314A7CD3CE22CB5D
8149314A7CD3C&FORM=VIRE
Everett Rogers (2003). Diffusion of Innovations, 5th
Edition.

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Five Stages in the Innovation-Decision Process

Rogers and Shoemaker (1973)

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