Chapter 8

Cost

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

Main Topics

Types of costWhat do economic costs include?Short-run cost: one variable inputLong-run cost: cost minimization with two

variable inputsAverage and marginal costsEffects of input price changesEconomies and diseconomies of scale

8-2

Types of Cost

Firm’s total cost is the expenditure required to produce a given level of output in the most economical way

Variable costs are the costs of inputs that vary with output level

Fixed costs do not vary as the level of output changes, although might not be incurred if production level is zeroAvoidable versus sunk costs

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Production Costs: An Example

Table 8.1: Fixed, Variable, and Total Costs of Producing Garden Benches

Number of Benches

Produced 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 2,000

132 $1,000 2,000 3,000

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Economic Costs

Some economic costs are hidden, such as lost opportunities to use inputs in other waysExample: Using time to run your own firm

means giving up the chance to earn a salary in another job

An opportunity cost is the cost associated with forgoing the opportunity to employ a resource in its best alternative use

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Short Run Cost:One Variable Input

If a firm uses two inputs in production, one is fixed 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 inputFirm’s total cost = variable cost + any fixed costs

Can be represented graphically or mathematically

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Figure 8.1: Variable Cost from Production Function

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Figure 8.2: Fixed, Variable, and Total Cost Curves

Dark red curve is variable cost

Green curve is fixed cost

Light red curve is total cost, vertical sum of VC and FC

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Long-Run Cost: Cost Minimization with Two Variable Inputs

In the long run, all inputs are variableFirm will have many efficient ways to

produce a given amount of output, using different input combinations

Which efficient combination is cheapest?Consider a firm with two variable inputs K

and L, and inputs and outputs that are finely divisible

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Isocost Lines

An isocost line connects all input combinations with the same 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), the negative of the ratio of input prices

CRKWL

LR

W

R

CK

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Isocost Lines, continued

Isocost lines closer to the origin represent lower total cost

A family of isocost lines contains, for given input prices, the isocost lines for all possible cost levels of the firm

Note the close relationship between isocost lines and consumer budget linesLines show bundles that have same costSlope is negative of the price ratio

8-11

Sample Problem 1:

Plot the isocost line for a total cost of $20,000 when the wage rate is $10 and the rental rate is $40.

How does the isocost line change if the wage rises to $20?

Least-Cost Production

How do we find the least-cost input combination 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 input combination does not overlap with the area above the Q-unit isoquant

Again, note the similarities to the consumer’s problem

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Garden Bench Example, Continued

In the long run, Naomi and Noah can vary the amount of garage space they rent and the number of workers they hire

An assembly worker earns $500 per week

Garage space rents for $1 per square foot per week

Inputs are finely divisible

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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 Assembly Workers, L

B

A

C = $3500

D

C = $3000

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Interior Solutions

A least-cost input combination that uses at least a little bit of every input is an interior solution

Interior solutions always satisfy the tangency condition: the isocost line is tangent to the isoquant thereOtherwise, the isocost line would cross the isoquantCreate an area of overlap between the area under

the isocost line and the area above the isoquantThis would not minimize the cost of production

8-16

Least-Cost Production and MRTS

Restate the tangency condition in terms of marginal products 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 equal across inputs when the firm is using a least-cost input combination

R

MP

W

MP

R

W

MP

MP KL

K

L or

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Least-Cost Input Combination

How can we find a firm’s least-cost input combination? If isoquant for desired level of output has declining

MRTS: Find an interior solution for which the tangency condition

formula holds That input combination satisfies the no-overlap rule and must

be the least-cost combination If isoquant does not have declining MRTS:

First identify interior combinations that satisfy the tangency condition, if any

Compare the costs of these combinations to the costs of any boundary solutions

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Sample Problem 2:

Suppose the production function for Gadget World is Q = 5L0.5K0.5. The wage rate is $25 and the rental rate is $50. What is the least-cost combination of producing 100 gadgets? 200?

The Firm’s Cost Function

To determine the firm’s cost function need to find least-cost input combination for every output level

Firm’s output expansion path shows the least-cost input combinations at all levels of output for fixed input prices

Firm’s total cost curve shows how total cost changes with output level, given fixed input prices

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Figure 8.10: Output Expansion Path and Total Cost Curve

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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

QQCQC

Q

CMC

8-22

Cost, Average Cost, andMarginal Cost

Table 8.3: Cost, Average Cost, and Marginal Cost for a Hypothetical Firm

Output (Q)

Tons per day

Total Cost (C)

(per day)

Marginal Cost

(per day)

Average Cost

(per day)

0 $0 $0 $0

1 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

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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 a

straight line connecting it to the originSlope of that line equals average costEfficient scale of production is the output level at

which AC is lowestMarginal cost:

Firm’s marginal cost of producing Q units of output is equal to the slope of its cost function at output level Q

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Figure 8.16: Relationship Between AC and MC

AC slopes downward where it lies above the MC curve

AC slopes upward where it lies below the MC curve

Where AC and MC cross, AC is neither rising nor falling

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Marginal Cost, Marginal Products, and Input Prices

Intuitively, a firm’s costs should be lower the more productive it is and the lower the input prices it faces

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

LK MP

W

MP

RMC

8-26

More Average Costs: Definitions

Apply idea of average cost to firm’s variable and fixed costs to find average variable cost and average fixed cost:

Since total cost is the sum of variable and fixed costs, average cost is the sum of AVC and AFC:

Q

VCAVC

Q

FCAFC

AFCAVCQ

FC

Q

VC

Q

FCVC

Q

CAC

8-27

Average Cost Curves

Fixed costs are constant so AFC is always downward sloping

At each level of output the AC curve is the vertical sum of the AVC and AFC curvesAverage cost curve lies above both AVC

and AFC at every output levelEfficient scale of production exceeds output

level where AVC is lowest

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Figure 8.18: AC, AVC, andAFC Curves

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Figure 8.20: AC, AVC, andMC Curves

8-30

Effects of Input Price Changes

Changes in input prices usually lead to changes in a firm’s least-cost production method

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 input and usually employs more

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

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Figure 8.21: Effect of an Input Price Change

Point A is optimal input mix when price of labor is four times more than the price of capital

Point B is optimal when labor and capital are equally costly

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Short-run vs. Long-run Costs

In the long run a firm can vary all inputs Will 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

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Figure 8.24: Input Response over the Long and Short Run

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Figure 8.25: Long-run and Short-run Costs

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

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Economies and Diseconomies of Scale

What are the implications of returns to scale?A firm experiences economies of scale when

its average cost falls as it produces moreCost rises less, proportionately, than the increase in

outputProduction technology has increasing returns to

scale

Diseconomies of scale occur when average cost rises with production

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Figure 8.28: Returns to Scale and Economies of Scale

8-38

Sample Problem 3 (8.12):

Noah and Naomi want to produce 100 garden

benches per week in two production plants. The

cost functions at the two plants are

and ,

and the corresponding marginal costs are MC1 =

600 – 6Q1 and MC2 = 650 – 4Q2. What is the best

output assignment between the two plants?

2111 3600 QQC 2

222 2650 QQC