Chap008 Cost Minimization

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Chapter 8

Cost

McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.



Main Topics

Types of cost

What do economic costs include?

Short-run cost: one variable inputLong-run cost: cost minimization with two

variable inputs

Average and marginal costsEffects of input price changes

Economies and diseconomies of scale

8-2



Types of Cost

Firm’s to tal cost is the expenditure required toproduce a given level of output in the mosteconomical way

Variable cos ts are the costs of inputs that vary

with output level

Fixed costs do not vary as the level of outputchanges, although might not be incurred if

production level is zero Avoidable versus sunk costs

Applications 8.1, 8.2, 8.3 for real world cost.8-3



Production Costs: An Example

Table 8.1: Fixed, Variable, and Total Costs of ProducingGarden Benches

Number of

BenchesProduced per

Week

Fixed Costs(per Week)

Variable Cost(per Week)

Total Cost(per Week)

0 $1,000 $0 $1,000

33 $1,000 500 1,500

74 $1,000 1,000 2,000132 $1,000 2,000 3,000

8-4



Economic Costs

Some economic costs are hidden, suchas lost opportunities to use inputs inother waysExample: Using time to run your own firm

means giving up the chance to earn a salaryin another job

An oppo r tun i ty cost is the costassociated with forgoing the opportunityto employ a resource in its bestalternative use

8-5



Short Run Cost:One Variable Input

If a firm uses two inputs in production, one isfixed in the short run

To determine the short-run cost function with

only one variable input:Identify the efficient method for producing a given

level of outputThis shows how much of the variable input to useFirm’s variable cost = cost of that amount of input

Firm’s total cost = variable cost + any fixed costs Can be represented graphically or

mathematically

8-6



Figure 8.1: Variable Cost fromProduction Function

8-7



Figure 8.2: Fixed, Variable, andTotal Cost Curves

Dark red curve isvariable cost

Green curve is fixedcost

Light red curve istotal cost, vertical

sum of VC and FC Applications 8.1, 8.2,

8.3 for real worldcost.

8-8



Long-Run Cost: Cost Minimizationwith Two Variable Inputs

In the long run, all inputs are variable

Firm will have many efficient ways to

produce a given amount of output, usingdifferent input combinations

Which efficient combination is cheapest?

Consider a firm with two variable inputs Kand L, and inputs and outputs that arefinely divisible

8-9



Isocost Lines

An isocost line connects all input combinations with thesame cost

If W is the cost of a unit of labor and R is the cost of a

unit of capital, the isocost line for total cost C is:

Rearranged,

Thus the slope of an isocost line is –(W/R), thenegative of the ratio of input prices

C RK WL

L R

W

R

C

K

8-10



Isocost Lines, continued

Isocost lines closer to the origin representlower total cost

Afam i ly o f isocost l ines

contains, for giveninput prices, the isocost lines for all possiblecost levels of the firm

Note the close relationship between isocost

lines and consumer budget linesLines show bundles that have same cost

Slope is negative of the price ratio

8-11



Least-Cost Production

How do we find the least-cost inputcombination for a given level of output?Find the lowest isocost line that touches the

isoquant for producing that level of outputNo-Overlap Rule: The area below the isocost

line that runs through the firm’s least-cost inputcombination does not overlap with the area

above the Q-unit isoquant Again, note the similarities to the consumer’s

problem

8-12



Garden Bench Example,Continued

In the long run, Naomi and Noah canvary the amount of garage space theyrent and the number of workers they hire

An assembly worker earns $500 per week

Garage space rents for $1 per square

foot per weekInputs are finely divisible

8-13



Figure 8.7: Least-Cost Method,No-Overlap Rule Example

Q = 140

Square Feetof Space, K

1 2 3 4 5 6

500

1000

1500

2000

2500

Number of AssemblyWorkers, L

B

A

C = $3500

D

C = $3000

8-14



Interior Solutions

A least-cost input combination that uses atleast a little bit of every input is an inter ior

solut ion

Interior solutions always satisfy the tangency condi t ion : the isocost line is tangent to theisoquant thereOtherwise, the isocost line would cross the isoquant

Create an area of overlap between the area under the isocost line and the area above the isoquant

This would not minimize the cost of production

8-15



Least-Cost Production and MRTS

Restate the tangency condition in terms of marginalproducts and input prices: Slope of isoquant = -(MRTSLK)

MRTS = ratio of marginal products Slope of isocost lines = -(W/R)

Thus the tangency condition says:

Marginal product per dollar spent must be equalacross inputs when the firm is using a least-cost inputcombination

R

MP

W

MP

R

W

MP

MP K L

K

L or

8-16



Least-Cost Input Combination

How can we find a firm’s least-cost input combination?

If isoquant for desired level of output has decliningMRTS:

Find an interior solution for which the tangency conditionformula holds

That input combination satisfies the no-overlap rule and mustbe the least-cost combination

If isoquant does not have declining MRTS:

First identify interior combinations that satisfy the tangencycondition, if any

Compare the costs of these combinations to the costs of anyboundary solutions

8-17



Application 8.4: Dell

Just-in-time manufacturing.

Manufacture when consumers want

Cut back on cost of inventory.Invest more in manufacturing and

information technology to speed up

production.Substitute manufacturing and information

technology for inventories.



The Firm’s Cost Function

To determine the firm’s cost function need to

find least-cost input combination for everyoutput level

Firm’s ou tput expansion path shows theleast-cost input combinations at all levels of output for fixed input prices

Firm’s to tal cost cu rve shows how total costchanges with output level, given fixed inputprices

8-19



Figure 8.10: Output ExpansionPath and Total Cost Curve

8-20



Average and Marginal Cost

A firm’s average cost , AC=C/Q, is its cost per unit of output produced

Marginal cost measures now much extra cost the

firm incurs to produce the marginal units of output, per unit of output added

As output increases:Marginal cost first falls and then rises

Average cost follows the same pattern

Q

QQC QC

Q

C MC

8-21



Cost, Average Cost, andMarginal Cost

Table 8.3: Cost, Average Cost, and Marginal Cost for aHypothetical Firm

Output (Q)

Tons per day

Total Cost (C)

(per day)

Marginal Cost

(per day)

Average Cost

(per day)

0 $0 $0 $01 1,000 1,000 1,000

2 1,800 800 900

3 2,100 300 700

4 2,500 400 625

5 3,000 500 600

6 3,600 600 600

7 4,300 700 614

8 5,600 1,300 700

8-22



AC and MC Curves

When output is finely divisible, can represent AC and MC as curves Average cost:

Pick any point on the total cost curve and draw astraight line connecting it to the originSlope of that line equals average costEfficient scale of production is the output level at

which AC is lowest

Marginal cost:Firm’s marginal cost of producing Q units of output

is equal to the slope of its cost function at outputlevel Q

8-23



Figure 8.16: RelationshipBetween AC and MC

AC slopes downwardwhere it lies abovethe MC curve

AC slopes upwardwhere it lies belowthe MC curve

Where AC and MCcross, AC is neither rising nor falling

8-24



Marginal Cost, Marginal Products,and Input Prices

Intuitively, a firm’s costs should be lower the

more productive it is and the lower the inputprices it faces

Formalize relationship between marginal cost,marginal products, and input prices using thetangency condition:

L K MP W

MP R MC

8-25



More Average Costs: Definitions

Apply idea of average cost to firm’s variable

and fixed costs to find average variable costand average fixed cost:

Since total cost is the sum of variable and

fixed costs, average cost is the sum of AVCand AFC:

Q

VC AVC

Q

FC AFC

AFC AVC Q

FC

Q

VC

Q

FC VC

Q

C AC

8-26



Average Cost Curves

Fixed costs are constant so AFC isalways downward sloping

At each level of output the AC curve isthe vertical sum of the AVC and AFCcurves Average cost curve lies above both AVC

and AFC at every output levelEfficient scale of production exceeds output

level where AVC is lowest

8-27



Figure 8.18: AC, AVC, and AFC Curves

8-28



Figure 8.20: AC, AVC, andMC Curves

8-29



Effects of Input Price Changes

Changes in input prices usually lead tochanges in a firm’s least-cost productionmethod

Responses to a Change in an Input Price: When the price of an input decreases, a firm’s least-

cost production method never uses less of that inputand usually employs more

For a price increase, a firm’s least-cost inputproduction method never uses more of that inputand usually employs less

8-30



Figure 8.21: Effect of an InputPrice Change

Point A is optimalinput mix when priceof labor is four times

more than the priceof capital

Point B is optimal

when labor andcapital are equallycostly

8-31



Short-run vs. Long-run Costs

In the long run a firm can vary all inputsWill choose least-cost input combination for each output level

In the short run a firm has at least one fixed input

Produce some level of output at least-cost input combination Can vary output from that in short run but will have higher costs than could achieve if all inputs were variable

Long-run average variable cost curve is the lower envelope of the short-run average cost curves

One short-run curve for each possible level of output

8-32



Figure 8.24: Input Response over the Long and Short Run

8-33



Figure 8.25: Long-run and Short-run Costs

8-34



Figure 8.26: Long-run and Short-run Average Cost Curves

8-35



Economies and Diseconomies of Scale

What are the implications of returns to scale?

A firm experiences economies of scale whenits average cost falls as it produces more

Cost rises less, proportionately, than the increase inoutput

Production technology has increasing returns toscale

Diseconomies of scale occur when averagecost rises with production

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Chap008 Cost Minimization