The Theory and Estimation of Production

• The Production Function• Short-Run Analysis of Total, Average,

and Marginal Product• Long-Run Production Function• Estimation of Production Functions• Importance of Production Functions

in Managerial Decision Making

Learning Objectives• Define production function and explain

difference between short-run and long-run production function

• Explain “law of diminishing returns”• Define the Three Stages of Production

and how it relates to the “law of diminishing returns”

• Describe different forms of production functions that are used.

• Briefly describe the Cobb-Douglas function

Importance of chapter

• To provide a framework for managerial decisions regarding allocation of firms resources

• Show how managers can determine which inputs and how much of each input to use to produce output efficiently

• This chapter serves as the foundation for later chapters, which describe in detail pricing and output techniques for managers interested in profit maximization

• Mathematically, the production function can be expressed as

Q=f( K, L)

• Q: level of output • K and L: inputs used in the production

process

• Key assumptions– Some given “state of the art” in the

production technology.– Whatever input or input combinations

are included in a particular function, the output resulting from their utilization is at the maximum level.

Short run vs. long run decisions

• In the short run some factors of production are fixed and this limits the choice in making input decisions

e.g. Car manufacturing company: Capital is fixed but labor and steel can be adjusted making them variable inputs

The short run production function is essentially a

function of only labor

• In the long run the manager can adjust all factors of production in the long run all inputs are variable.

• If it takes a company 3 years to acquire additional capital machines, then the long run for that company is 3 years and the short run is less than 3 years

In summary:• The short-run production function

shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the amount of at least one of the inputs used remains unchanged.

• The long-run production function shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the firm is free to vary the amount of all the inputs being used.

Short run functions• Assume Q = F(K,L) = K.5 L.5

– K is fixed at 16 units. – Short run production function:

Q = (16).5 L.5 = 4 L.5

– Production when 100 units of labor are used?

Q = 4 (100).5 = 4(10) = 40 units

Measures of productivity

• Managers must determine the productivity of inputs used in the production process

• This is useful for evaluating the effectiveness of the production process and making input decisions that maximize profit

• 3 most important measures of productivity are Total product, Average product and Marginal product

Short-Run Analysis of Total,Average, and Marginal Product

• Alternative terms in reference to inputs– Inputs– Factors– Factors of production– Resources

• Alternative terms in reference to outputs– Output– Quantity (Q)– Total product (TP)– Product

Average Product:• Manager may wish to know, on

average, how much each worker contributes to the total output of the firm.

• AP for an input is • Total product divided by quantity

use of input

• Average Product of Labor– APL = Q/L.– Measures the output of an “average”

worker.– Example: Q = F(K,L) = K.5 L.5

• If the inputs are K = 16 and L = 9, then the average product of labor is APL = [(16)

• 0.5(9)0.5]/9 = 1.33

Marginal Product:• Is the change in total output

attributable to the last unit of input

• MP for an input is • Change in Total product divided by

change in quantity use of input

Average Product (AP):

X

QMPX

X

QAPX

•Marginal product (MP):

• If MP > AP then AP is rising.

• If MP < AP then AP is falling.

• MP=AP when AP is maximized.

Phases of Marginal Product:• As the usage of an input

increases, marginal product initially increases (increasing marginal returns), then begins to decline (decreasing marginal returns) and eventually becomes negative (negative marginal returns)

Q

L

Q=F(K,L)

IncreasingMarginalReturns

DiminishingMarginalReturns

NegativeMarginalReturns

MP

AP

Increasing, Diminishing and Negative Marginal Returns

• Law of Diminishing Returns: As additional units of a variable input are combined with a fixed input, at some point the additional output (i.e., marginal product) starts to diminish.– Nothing says when diminishing returns will start

to take effect, only that it will happen at some point.

– All inputs added to the production process are exactly the same in individual productivity

• The Three Stages of Production in the Short Run– Stage I: From zero units of the

variable input to where AP is maximized (where MP=AP)

– Stage II: From the maximum AP to where MP=0

– Stage III: From where MP=0 on

• In the short run, rational firms should only be operating in Stage II.

• Why not Stage III?– Firm uses more variable inputs to produce less

output

• Why not Stage I?– Underutilizing fixed capacity– Can increase output per unit by increasing the

amount of the variable input

What level of input usage within Stage II is best for the firm?

The answer depends upon how many units of output the firm can sell, the price of the product, and the monetary costs of employing the variable input.

Labor is hired for a given K until the additional revenue (Marginal revenue of labor) equals the marginal cost of labor (wage)

Determining the right level of input:

e.g. restaurant manager must hire the “correct” number of servers

If product is sold at $3 on the market and each unit of labor costs $400, how many units of labor should be hired to maximize profit?

• First, determine the benefit of hiring an additional worker. Each worker increases the firm’s total output by her marginal product.

• This increase can be sold in a market at a price of $3

• Thus the benefit from each unit of labor is $3 x MP of worker

• This number is known as the Value marginal product of labor = VMP

• VMPL = P x MPL.

• It is profitable to hire units of labor so long as their additional output value exceeds their cost.

• So, employ labor as long as VMP exceeds their wage (w)

• To maximize profits, a manager should use inputs at levels which their marginal benefits equal the marginal cost.

• Specifically for labor, • VMPL = w

• For capital: capital is rented up to the point where

value of marginal product of capital equals the rental rate: VMPK = r,

• Summary of relationship between demand for output and demand for input– A profit-maximizing firm operating in

perfectly competitive output and input markets will be using the optimal amount of an input at the point at which the monetary value of the input’s marginal product is equal to the additional cost of using that input.

– MRP = MLC

Size (sq ft)

Net Revenue (‘000 of $)

20,0000 165

265 347 420 480 520 550 575

15,0000 145

235 300 350 390 420 442 460

10,0000 120

190 255 315 345 365 380 390

50,000 93 135 180 210 235 255 273 288

L(workers) 10 20 30 40 50 60 70 80If the wage per worker is $3,000

1. How many workers should a 50,000 sq ft store employ?2. How many workers should a 200,000 sq ft store employ?

• Multiple variable inputs– Consider the relationship between the ratio of

the marginal product of one input and its cost to the ratio of the marginal product of the other input(s) and their cost.

– Other factors may outweigh this relationship• Political/Economic risk factors`

k

k

w

MP

w

MP

w

MP

2

2

1

1

Case 1 Case 2 Case 3

Case 4 Case 5

A B A B A B A B A B

Output of task 1 5 4 5 5 5 5 5 1 5 1

Output of task 2 10 2 10 3 10

5 10 1 10

5

Wage ($) 15 6 15 6 15

6 15 6 15

6

Workers A and B can perform task 1 and/or task 2.The table shows their output (in units) per hour and their hourly wage rate (w)

If worker A and B can flexibly work with the same fixed stock of capital (equivalent to ignoring capital), who should perform task 1 and/ or task 2. (perform this exercise for each case)

The Long-Run Production Function

• In the long run, a firm has enough time to change the amount of all its inputs.– Effectively, all inputs are variable.

• The long run production process is described by the concept of returns to scale.

• If all inputs into the production process are doubled, three things can happen:– output can more than double

• increasing returns to scale (IRTS)

– output can exactly double• constant returns to scale (CRTS)

– output can less than double• decreasing returns to scale (DRTS)

• One way to measure returns to scale is to use a coefficient of output elasticity:

• If EQ > 1 then IRTS• If EQ = 1 then CRTS• If EQ < 1 then DRTS

inputsallinchangePercentage

QinchangePercentageQE

• Returns to scale can also be described using the following equation

hQ = f(kX, kY)

• If h > k then IRTS• If h = k then CRTS• If h < k then DRTS

• Graphically, the returns to scale concept can be illustrated using the following graphs.

Q

X,Y

IRTSQ

X,Y

CRTSQ

X,Y

DRTS

Estimation of Production Functions

• Forms of Production Functions– Cobb-Douglas Production Function: Q = aLbKc

• Both capital and labor inputs must exist for Q to be a positive number

• Can be increasing, decreasing, or constant returns to scale

– b + c > 1, IRTS– b + c = 1, CRTS– b + c < 1, DRTS

• Permits us to investigate MP for any factor while holding all others constant

• Elasticities of factors are equal to their exponents

Estimation of Production Functions

• Forms of Production Functions– Cobb-Douglas Production Function

• Can be estimated by linear regression analysis• Can accommodate any number of independent variables• Does not require that technology be held constant• Shortcomings:

– Cannot show MP going through all three stages in one specification

– Cannot show a firm or industry passing through increasing, constant, and decreasing returns to scale

– Specification of data to be used in empirical estimates`

Marginal Rate of Technical Substitution

(MRTS)

• The rate at which two inputs are substituted while maintaining the same output level.

K

LKL MP

MPMRTS

Cost Minimization

• Marginal product per dollar spent should be equal for all inputs:

• But, this is justr

w

MP

MP

r

MP

w

MP

K

LKL

r

wMRTSKL

The Demand for Labor

• Labor is hired for a given K until the additional revenue (Marginal revenue of labor) equals the marginal cost of labor (wage)

• Demand for labor (or for any factor of production) is a derived demand.

• It is derived from the demand for the final product

• The demand for labor is affected by

1.Changes in output2.Use of capital3.Technology4.Price of substitute inputs5.Price of complement inputs6.Price of the final product

The demand for an input will be more elastic the

1. Higher the sensitivity of the final product to changes in price

2. Higher the share of the input in the buyer’s total cost

3. Higher the availability of alternative inputs

4. The higher the substitutability of complement inputs

Factors behind the demand for inputs

Case 1Price of wood increases price of furniture

increases quantity of furniture decreases considerably if the demand for furniture is elastic quantity of wood decreases considerably.

Case 2Price of beef increases price of beef soup

increases quantity of soup decreases considerably if beef has a high share in the cost of beef soup

Case 3Price of beef increases demand for substitute

chicken increases quantity of chicken increases considerably if supply of chicken is elastic quantity of beef decreases considerably

Case 4Price of tires decrease demand for

complement input (wheel covers) increases substitutes for complement input are readily available quantity of tires increases considerably

Supply of Labor

In a normal case a worker supplies more hours of work as the wage increases and the supply of labor is upward sloping

The incentive to work more hours outweighs the higher demand for leisure as income rises

Supply of labor

• In a special case a worker supplies fewer hours of work when the wage increases and the supply curve is backward bending

• The higher demand for leisure as income rises outweighs the incentive to work more hours

Remuneration (Pay) and Reservation Wage

Job A: Pay (benefits) – costs = net benefits of AJob B: Pay (benefits) – costs = net benefits of B

Instead of Job A, a person can choose B, the best feasible alternative

Gain from Job a is the incremental net benefit of Job A relative to the net benefit of Job B

Gain of A = (Pay in A – Cost of A) – (Pay in B – Cost in B)

• Economic gain of A is(Pay in A – Cost of A) – Opportunity cost of A

Reservation wage is the minimum a worker is willing to accept to work in A. The reservation wage is then Costs in A + (Pay in B – Costs of B)

1. What is the effect (on employment and output) of adopting Labor-saving technology?

2. Do cheaper (lower rental rate) machines reduce employment?

The Theory and Estimation of Cost

• The Importance of Cost in Managerial Decisions• The Definition and Use of Cost in Economic

Analysis• The Relationship Between Production and Cost• The Short Run Cost Function• The Long Run Cost Function• The Learning Curve• Economies of Scope• Economies of Scale: the Short Run Versus the

Long Run• Supply Chain Management• Ways Companies Have Cut Costs to Remain

Competitive

The Importance of Costin Managerial Decisions

• Ways to contain or cut costs over the past decade– Most common: reduce number of

people on the payroll– Outsourcing components of the

business– Merge, consolidate, then reduce

headcount

Cost Topology

Opportunity costs I(implicit or economic costs) and explicit cost (accounting costs)

Total cost (TC) = Fixed Costs (TFC) + Variable Costs (TVC)

Fixed costs do not change with output. Variable cost change with output.

This distinction is only valid in the S/R

• Relevant cost: a cost that is affected by a management decision.

• Incremental cost: additional cost due to additional units of output

• Sunk cost: does not vary in accordance with decision alternatives. Are not recoverable and should not be considered

The Relationship Between Production and Cost

• Cost function is simply the production function expressed in monetary rather than physical units.

• Assume the firm is a “price taker” in the input market.

The Relationship Between Production and Cost

• Total Variable Cost (TVC): the cost associated with the variable input, determined by multiplying the number of units by the unit price.

• Marginal Cost (MC): the rate of change in total variable cost.

• The law of diminishing returns implies that MC will eventually increase

MP

W

Q

TVCMC

The Relationship Between

Production and Cost• Plotting TP and

TVC illustrates that they are mirror images of each other.

• When TP increases at an increasing rate, TVC increases at a decreasing rate.

The Short-Run Cost Function

• Standard variables in the short-run cost function:– Quantity (Q): the amount of output that a

firm can produce in the short run.– Total fixed cost (TFC): the total cost of using

the fixed input, capital (K)– Total variable cost (TVC): the total cost of

using the variable input, labor (L)– Total cost (TC): the total cost of using all the

firm’s inputs, L and K.TC = TFC + TVC

Total and Variable Costs

TC(Q): Minimum total cost of producing alternative levels of output:

TC(Q) = TVC(Q) + TFC

TVC(Q): Costs that vary with output.

TFC: Costs that do not vary with output.

$

Q

C(Q) = VC + FC

VC(Q)

FC

0

Fixed and Sunk Costs

FC: Costs that do not change as output changes.

Sunk Cost: A cost that is forever lost after it has been paid.

$

Q

FC

C(Q) = VC + FC

VC(Q)

The Short-Run Cost Function

• Standard variables in the short-run cost function:– Average fixed cost (AFC): the average per-unit cost of

using the fixed input K.AFC = TFC/Q

– Average variable cost (AVC): the average per-unit cost of using the variable input L.AVC = TVC/Q

– Average total cost (AC) is the average per-unit cost of using all the firm’s inputs.AC = AFC + AVC = TC/Q

– Marginal cost (MC): the change in a firm’s total cost (or total variable cost) resulting from a unit change in output.MC = TC/Q = TVC/Q

Average Total CostATC = AVC + AFCATC = C(Q)/Q

Average Variable CostAVC = VC(Q)/Q

Average Fixed CostAFC = FC/Q

Marginal CostMC = C/Q

$

Q

ATCAVC

AFC

MC

MR

The Short-Run Cost Function

• Important Observations– AFC declines steadily over the range of

production.– When MC = AVC, AVC is at a minimum.– When MC < AVC, AVC is falling.– When MC > AVC, AVC is rising.– The same three rules apply for average

cost (AC) as for AVC.

The Short-Run Cost Function

• A reduction in the firm’s fixed cost would cause the average cost line to shift downward.

• A reduction in the firm’s variable cost would cause all three cost lines (AC, AVC, MC) to shift.

The Short-Run Cost Function

• Alternative specifications of the Total Cost function– Most commonly: specified as a cubic

relationship between total cost and output• As output increases, total cost first

increases at a decreasing rate, then increases at an increasing rate.

• TC = a + bQ + cQ2 + dQ3

The Short-Run Cost Function

• .

– Quadratic relationship• As output increases, total cost increases

at an increasing rate.• TC = a + bQ + cQ2

– Linear relationship• As output increases, total cost increases

at a constant rate.TC = a + bQ

Cubic Cost Function• C(Q) = f + a Q + b Q2 + cQ3

• Marginal Cost?dC/dQ = a + 2bQ + 3cQ2

An Example– Total Cost: C(Q) = 10 + Q + Q2

– Variable cost function:VC(Q) = Q + Q2

– Variable cost of producing 2 units:VC(2) = 2 + (2)2 = 6

– Fixed costs:FC = 10

– Marginal cost function:MC(Q) = 1 + 2Q

– Marginal cost of producing 2 units:MC(2) = 1 + 2(2) = 5

The Long-Run Cost Function

• In the long run, all inputs to a firm’s production function may be changed.

• Because there are no fixed inputs, there are no fixed costs.

• The firm’s long run marginal cost pertains to returns to scale.– First, increasing returns to scale.– As firms mature, they achieve constant returns,

then ultimately decreasing returns to scale.

The Long-Run Cost Function

• When a firm experiences increasing returns to scale:– A proportional increase in all inputs

increases output by a greater proportion.

– As output increases by some percentage, total cost of production increases by some lesser percentage.

The Long-Run Cost Function

• Economies of Scale: situation where a firm’s long-run average cost (LRAC) declines as output increases.

• Diseconomies of Scale: situation where a firm’s LRAC increases as output increases.

• In general, the LRAC curve is u-shaped.

Economies of Scale

LRAC

$

Q

Economiesof Scale

Diseconomiesof Scale

Reasons for long-run economies

– Specialization in the use of labor and capital.

– Prices of inputs may fall as the firm realizes volume discounts in its purchasing.

– Use of capital equipment with better price-performance ratios.

– Larger firms may be able to raise funds in capital markets at a lower cost than smaller firms.

– Management efficiencies (fewer people run more operations)

Reasons for Diseconomies of Scale

• Reasons for Diseconomies of Scale– Scale of production becomes so large

that it affects the total market demand for inputs, so input prices rise.

– Transportation costs tend to rise as production grows.• Handling expenses, insurance, security, and

inventory costs affect transportation costs.

Why economies are important

• Economies of scale protect existing firms from entrants by allowing for low average costs at high output levels.

• Barriers to entry sustain profits

The Long-Run Cost Function

• In long run, the firm can choose any level of capacity.

• Once it commits to a level of capacity, at least one of the inputs must be fixed. This then becomes a short-run problem.

• The LRAC curve is an envelope of SRAC curves, and outlines the lowest per-unit costs the firm will incur over a range of output.

Economies of Scope• Economies of Scope: reduction of

a firm’s unit cost by producing two or more goods or services jointly rather than separately.

• Closely related to economies of scale.

Multi-Product Cost Function

• C(Q1, Q2): Cost of jointly producing two outputs.

• General function form:

22

212121, cQbQQaQfQQC

Economies of Scope• TC(Q1, 0) + TC(0, Q2) > TC(Q1, Q2).

– It is cheaper to produce the two outputs jointly instead of separately.

• Example:– It is cheaper for Time-Warner to

produce Internet connections and Instant Messaging services jointly than separately.

Cost Complementarity• The marginal cost of producing good 1

declines as more of good two is produced:

MC1Q1,Q2) /Q2 < 0.

• Example:– Cow hides and steaks.

Quadratic Multi-Product Cost Function

• TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2

• MC1(Q1, Q2) = aQ2 + 2Q1

• MC2(Q1, Q2) = aQ1 + 2Q2

• Cost complementarity: a < 0

• Economies of scope: f > aQ1Q2

TC(Q1 ,0) + TC(0, Q2 ) = f + (Q1 )2 + f + (Q2)2

TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2

f > aQ1Q2: Joint production is cheaper

A Numerical Example:• TC(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2

• Cost Complementarity?Yes, since a = -2 < 0MC1(Q1, Q2) = -2Q2 + 2Q1

• Economies of Scope?Yes, since 90 > -2Q1Q2

Production Costs, Organization Costs, and Transaction Costs as

Determinants of Optimum Firm Size

• The Firm starts small and benefits from economies of scale and scope in production, horizontal integration, and multi-plant production as it grows

• The firm continues growing by integrating vertically with suppliers, downstream producers and intermediate customers. The firm reduces transaction costs, replacing higher transaction costs with lower internal organizational costs

• However as the firm gets larger, increasing internal organization costs outweigh the economies of scale and scope in production, the benefits from horizontal integration and multi-plant production, and the lower transactions costs

•

Economies of Scale and Firm Size

Profits do not depend only on economies of scale in production

–Diseconomies of scale in areas other than production may outweigh economies of scale in production–Smaller producers may benefit from niche markets–Smaller companies may be in better position to “read” the customer and innovate–As size increases, organizational (agency) costs rise