Production Theory and Cost1

Production and Costs

2

Figure 2: The Firm’s Production Function

Alternative Input

Combinations

Different Quantities of Output

Production Function

Production Function

The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (K) and labor (L)

qq = = ff((KK,,LL))

Time Horizon:The Short Run and the Long Run

Useful to categorize firms’ decisions into –Long-run decisions—involves a time

horizon long enough for a firm to vary all of its inputs

–Short-run decisions—involves any time horizon over which at least one of the firm’s inputs cannot be varied

Production in the Short Run• There is nothing they can do about their fixed inputs

– Stuck with whatever quantity they have– However, can make choices about their variable inputs

• Fixed inputs– An input whose quantity must remain constant, regardless

of how much output is produced

• Variable input– An input whose usage can change as the level of output

changes

Production in the Short Run

• Total product– Maximum quantity of output that can be produced

from a given combination of inputs• Marginal product of labor (MPL) is the change in total

product (ΔQ) divided by the change in the number of workers hired (ΔL)

ΔL

ΔQMPL

– Tells us the rise in output produced when one more worker is hired

Total and Marginal Product

30

90

130161184196 Total Product

Q from hiring fourth worker

Q from hiring third worker

Q from hiring second worker

Q from hiring first worker

increasing marginal returns

diminishing marginal returns

Units of Output

Number of Workers62 3 4 51

Marginal Returns To Labor

• As more and more workers are hired– MPL first increases– Then decreases

• Pattern is believed to be typical at many types of firms

Increasing Marginal Returns to Labor

• When the marginal product of labour increases as employment rises, we say there are increasing marginal returns to labor– Each time a worker is hired, total output

rises by more than it did when the previous worker was hired

Diminishing Returns To Labour

• When the marginal product of labor is decreasing – There are diminishing marginal returns to labour– Output rises when another worker is added so marginal

product is positive– But the rise in output is smaller and smaller with each

successive worker• Law of diminishing (marginal) returns states that as we

continue to add more of any one input (holding the other inputs constant)– Its marginal product will eventually decline

Marginal Product

To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant

KK fK

qMP

capital ofproduct marginal

LL fL

qMP

labor ofproduct marginal

Diminishing Marginal Productivity

• The marginal physical product of an input depends on how much of that input is used

• In general, we assume diminishing marginal productivity

02

KK

K fK

q

K

MP

02

LL

L fL

q

L

MP

Average Product

• Labor productivity is often measured by average productivity

L

LKf

L

qAPL

),(

input labor

output

Note that Note that APAPLL also depends on the amount of also depends on the amount of capital employedcapital employed

Total Product, Average Product, and Marginal Product Curves

A Two-Input Production Function

• Suppose the production function for flyswatters can be represented by

qq = = ff((KK,,LL) = 600) = 600KK 22LL22 - - KK 33LL33

• To construct MPL and APL, we must assume a value for K – Let K = 10

• The production function becomesqq = 60,000 = 60,000LL22 - 1000 - 1000LL33


• The marginal productivity function is MPMPLL = = qq//LL = 120,000 = 120,000LL - 3000 - 3000LL22

which diminishes as L increases• This implies that q has a maximum value:

120,000120,000LL - 3000 - 3000LL22 = 0 = 04040LL = = LL22

LL = 40 = 40• Labor input beyond L=40 reduces output


• To find average productivity, we hold K=10 and solve

APAPLL = = qq//LL = 60,000 = 60,000LL - 1000 - 1000LL22

• APL reaches its maximum where

APAPLL//LL = 60,000 - 2000 = 60,000 - 2000LL = 0 = 0

LL = 30 = 30


• In fact, when L=30, both APL and MPL are equal to 900,000

• Thus, when APL is at its maximum, APL and MPL are equal

Isoquant Maps

• To illustrate the possible substitution of one input for another, we use an isoquant map.

• An isoquant shows those combinations of K and L that can produce a given level of output (q0)

ff((KK,,LL) = ) = qq00

Isoquant Map

L per period

K per period

Each isoquant represents a different level of outputoutput rises as we move northeast

q = 30q = 20

Marginal Rate of Technical Substitution (MRTS)

L per period

K per period

q = 20

- slope = marginal rate of technical substitution (MRTS)

The slope of an isoquant shows the rate at which L can be substituted for K

LA

KA

KB

LB

A

B

MRTS > 0 and is diminishing forincreasing inputs of labor

Returns to Scale

• How does output respond to increases in all inputs together?

• Suppose that all inputs are doubled, would output double?

• Returns to scale have been of interest to economists since the days of Adam Smith

Returns to Scale

• If the production function is given by q = f(K,L) and all inputs are multiplied by the same positive constant (m > 1), then

Effect on Output Returns to Scale

f(mK,mL) = mf(K,L) Constant

f(mK,mL) < mf(K,L) Decreasing

f(mK,mL) > mf(K,L) Increasing

The Linear Production Function

• Suppose that the production function isqq = = ff((KK,,LL) = ) = aKaK + + bLbL

• This production function exhibits constant returns to scale

ff((mKmK,,mLmL) = ) = amKamK + + bmLbmL = = mm((aKaK + + bLbL) = ) = mfmf((KK,,LL))

• All isoquants are straight linesAll isoquants are straight lines– MRTSMRTS is constant is constant

The Linear Production Function

L per period

K per period

q1q2 q3

Capital and labor are perfect substitutes

MRTS is constant as K/L changes

slope = -b/a

Fixed Proportions

L per period

K per period

q1

q2

q3

No substitution between labor and capital is possible

K/L is fixed at b/a

q3/b

q3/a

Cobb-Douglas Production Function

• Suppose that the production function isqq = = ff((KK,,LL) = ) = AKAKaaLLbb A,a,b A,a,b > 0> 0

• This production function can exhibit any This production function can exhibit any returns to scalereturns to scale

ff((mKmK,,mLmL) = ) = AA((mKmK))aa((mLmL)) bb = = AmAmaa++b b KKaaLLbb = = mmaa++bbff((KK,,LL))– if if aa + + bb = 1 = 1 constant returns to scale constant returns to scale– if if aa + + bb > 1 > 1 increasing returns to scale increasing returns to scale– if if aa + + bb < 1 < 1 decreasing returns to scale decreasing returns to scale


• Suppose that hamburgers are produced according to the Cobb-Douglas function

qq = 10 = 10KK 0.5 0.5 LL0.50.5

• Since a+b=1 constant returns to scale• The isoquant map can be derived

qq = 50 = 10 = 50 = 10KK 0.5 0.5 LL0.5 0.5 KLKL = 25 = 25qq = 100 = 10 = 100 = 10KK 0.5 0.5 LL0.5 0.5 KLKL = 100 = 100

– The isoquants are rectangular hyperbolasThe isoquants are rectangular hyperbolas


• The MRTS can easily be calculated

L

K

KL

KL

f

fKLMRTS

K

L

5.05.0

5.05.0

5

5)for (

The The MRTSMRTS declines as declines as LL rises and rises and KK falls falls The The MRTSMRTS depends only on the ratio of depends only on the ratio of KK and and

LL

Costs• A firm’s total cost of producing a given level of

output is the opportunity cost of the owners– Explicit (involving actual payments)

• Money actually paid out for the use of inputs– Implicit (no money changes hands)

• The cost of inputs for which there is no direct money payment

• This is the core of economists’ thinking about costs

Economic Profit vs Accounting Profit

• Accounting profit– The business’s revenue minus the explicit cost

and depreciation• Depreciation occurs because machines war

out over time• Economic profit

– The business’s revenue minus opportunity cost• In economics, profit is simplification of economic

profit.

Economic Cost vs Accounting Cost

• Accounting cost: Explicit costs - actual expenses and depreciation

• Economic cost: Opportunity cost – costs associated with opportunities that are foregone

Sunk Cost

• A sunk cost is an expenditure that has already been made and cannot be recovered.

• No influence on firm’s decisions• No alternative use• Zero opportunity cost

Costs in the Short Run

• Fixed costs – Costs of a firm’s fixed inputs

• Variable costs – Costs of obtaining the firm’s variable inputs

Measuring Short Run Costs: Total Costs

• Types of total costs– Total fixed costs

• Cost of all inputs that are fixed in the short run– Total variable costs

• Cost of all variable inputs used in producing a particular level of output

– Total cost• Cost of all inputs—fixed and variable• TC = TFC + TVC

Measuring Short Run Costs: Total Costs

Total cost (TC) is a function of output (Q)TC = f(Q)TC = TFC + TVC

+ L.wK

The Firm’s Total Cost Curves In The Short Run

TC

0

Dollars

135

195

255

315

375

$435

30 90 130 161

Units of Output

184 196

TFC

TFC

TVC

Average Costs

• Average fixed cost (AFC)– Total fixed cost per unit of output produced

• Average variable cost (TVC)– Total variable cost per unit of output produced

• Average total cost (TC)– Total cost per unit of output produced

Q

TFCAFC

Q

TVCAVC

Q

TCATC

Marginal Cost

• Marginal Cost– Increase in total cost from producing one more

unit or output• Marginal cost is the change in total cost (ΔTC) divided

by the change in output (ΔQ)

ΔQ

ΔTCMC

– Tells us how much cost rises per unit increase in output– Marginal cost for any change in output is equal to shape

of total cost curve along that interval of output

Average And Marginal Costs In The Short Run

MC

AVCATCAFC

Units of Output

Dollars

$4

3

2

1

30 90 130 161 1960

The Shape of the Marginal Cost Curve

• When the marginal product of labor (MPL) rises (falls), marginal cost (MC) falls (rises)

• Since MPL ordinarily rises and then falls, MC will do the opposite—it will fall and then rise–Thus, the MC curve is U-shaped

The Relationship Between Average And Marginal Costs

• At low levels of output, the MC curve lies below the AVC and ATC curves– These curves will slope downward

• At higher levels of output, the MC curve will rise above the AVC and ATC curves– These curves will slope upward

• As output increases; the average curves will first slope downward and then slope upward– Will have a U-shape

• MC curve will intersect the minimum points of the AVC and ATC curves

Production And Cost in the Long Run

• Goal: earn the highest possible profit– To do this, it must follow the least cost rule

• To produce any given level of output the firm will choose the input mix with the lowest cost

• Firm must decide what combination of inputs to use in producing any level of output

• Long-run total cost (LRTC)– The cost of producing each quantity of output when the least-cost

input mix is chosen in the long run• Long-run average total cost (LRATC)

– The cost per unit of output in the long run, when all inputs are variable

Q

LRTCLRATC

The Relationship Between Long-Run And Short-Run Costs

• For some output levels, LRTC is smaller than TC• Long-run total cost of producing a given level of

output can be less than or equal to, but never greater than, short-run total cost (LRTC ≤ TC)

• Long-run average cost of producing a given level of output can be less than or equal to, but never greater than, short–run average total cost (LRATC ≤ ATC)

Plant Size

• Plant - collection of fixed inputs at a firm’s disposal

• Can distinguish between the long run and the short run– In the long run, the firm can change the size

of its plant– In the short run, it is stuck with its current

plant size

Average Cost And Plant Size• ATC curve tells us how average cost behaves in the

short run, given plant size fixed– moving along the current ATC curve

• To produce any level of output in the long run, the firm will always choose that ATC curve with lowest ATC —among all of the ATC curves available– move from one ATC curve to another by varying the size

of its plant– Will also be moving along its LRATC curve– This insight tells us how we can graph the firm’s LRATC

curve

Long-Run Average Total CostFor each output level, firm will always choose to operate

on the ATC curve with the lowest possible cost

LRATCATC1

Use 0 automated

lines

ATC3ATC0

C

BA

ATC2

D

E

175 196184

Dollars

1.00

2.00

3.00

$4.00

Units of Output

30 90 130 161 250 3000

Use 1 automated

lines

Use 2 automated

lines

Use 3 automated

lines

Economics of Scale

• According to whether the LRATC decreases / does not change / increase as output increases, there are three types of issues:– Economies of scale (decreasing LRATC) at

relatively low levels of output– Constant returns to scale (constant LRATC) at

some intermediate levels of output– Diseconomies of scale (increasing LRATC) at

relatively high levels of output• LRATC curves are typically U-shaped

The Shape Of LRATC

Units of Output

LRATC

Economies of Scale Constant Returns to

Scale

Diseconomies of Scale

Dollars

1.00

2.00

3.00

$4.00

130 1840

Economies of Scale• An increase in output causes LRATC to decrease

– The more output produced, the lower the cost per unit

– LRATC curve slopes downward– Long-run total cost rises proportionately less

than output– Increasing return to scale

Why Should A Firm Experience The Economies of Scale?

• Gains from specialization

– greatest opportunities for increased specialization at a relatively low level of output

• More efficient use of lumpy inputs– Some types of inputs cannot be increased in tiny

increments, but rather must be increased in large jumps, therefore must be purchased in large lumps

• Low cost per unit is achieved only at high levels of output

• More efficient use of lumpy inputs will have more impact on LRATC at low levels of outputs

Diseconomies of Scale

• LRATC increases as output increases– LRATC curve slopes upward– LRTC rises more than in proportion to output– More likely at higher output levels

• As output continues to increase, most firms will reach a point where bigness begins to cause problems– True even in the long run, when the firm is free to

increase its plant size as well as its workforce

Production Theory and Cost1

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