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Chapter 8: Cost Curves

- A firm aims to MAXIMIZE PROFITS
- In order to do this, one must understand how to MINIMIZE COSTS
- Therefore understanding of cost curves is essential to maximizing profits

Chapter 8: Costs Curves

In this chapter we will cover:

8.1 Long Run Cost Curves

8.1.1 Total Cost

8.1.2 Marginal Cost and Average Cost

8.2 Economies of Scale

8.3 Short Run Cost Curves

8.3.1 Total Cost, Variable Cost, Fixed Cost

8.3.2 Marginal Cost and Average Cost

8.4 Economies of Scope

8.5 Economies of Experience

8.1 Long Run Cost Curves

- In the long run, a firm’s costs equal zero when zero production is undertaken
- As production (Q) increases, the firm must use more inputs, thus increasing its cost
- By minimizing costs, a firm’s long run cost curve is as follows:

Q1

Q0

•

TC = TC0

K1

•

K0

TC = TC1

0

L (labor services per year)

L0

L1

TC ($/yr)

LR Total Cost Curve

TC1=wL1+rK1

TC0 =wL0+rK0

Q (units per year)

Q1

0

Q0

Input Prices and LR Cost Curves

- An increase in the price of only 1 input will cause a firm to change its optimal choice of inputs
- However, the increase in input costs will always cause a firm’s costs to increase:
- -(This is only not true in the case of perfect substitutes when the productivity per dollar of each substitute is originally equal)

C1: Original isocost curve

(TC = $200)

C2: Isocost curve after

Price change (TC = $200)

C3: Isocost curve after

Price change (TC = $300)

Slope=w1/r

TC1/r

A

•

TC0/r

B

•

Slope=w2/r

Q0

C2

C3

C1

0

L

Change in Input Prices ->A Shift in the Total Cost Curve

TC(Q) new

TC(Q) old

300

200

Q (units/yr)

Q0

- Let Q=2(LK)1/2 MRTS=K/L,
- W=5, R=20, Q=40
- What occurs to costs when rent falls to 5?
- Initially:
- MRTS=W/R
- K/L=5/20
- 4K=L

- Q=2(LK)1/2
- 40=2(4KK)1/2
- 40=4K
- 10=K
- 40=L

- Let Q=2(LK)1/2 MRTS=K/L,
- W=5, R=20, Q=40
- What occurs to costs when rent falls to 5?
- After Price Change:
- MRTS=W/R
- K/L=5/5
- L=K

- Q=2(LK)1/2
- 40=2(LL)1/2
- 40=2L
- 20=L
- 20=K

What occurs when rent falls to 5?

- Initial: L=40, K=10 Final: L=K=20
- W=5, R=20, Q=40
- Initial: TC=wL+rK
- TC=5(40)+20(10)
- TC=400
- Final: TC=5(20)+5(20)
- TC=200
- Due to the fall in rent, total cost falls by $200.

- To calculate total cost, simply substitute labour and capital demand into your cost expression:
- Q= 50L1/2K1/2 (From Chapter 7, slide 38:)
- L*(Q,w,r) = (Q0/50)(r/w)1/2
- K*(Q,w,r) = (Q0/50)(w/r)1/2
- TC = wL +rK
- TC= w [(Q0/50)(r/w)1/2 ]+r[(Q0/50)(w/r)1/2 ]
- TC= [(Q0/50)(wr)1/2 ]+[(Q0/50)(wr)1/2 ]
- TC = 2Q0(wr)1/2 /50

Let Q= L1/2K1/2, MPL/MPK=K/L, w=10, r=40. Calculate total cost.

MRTS=w/r

K/L=10/40

K=4L

Q=L1/2K1/2 =L1/2(4L)1/2

Q=2L

L=Q/2

Let Q= L1/2K1/2, MRTS=K/L, w=10, r=40.

Calculate total cost.

Q=L1/2K1/2 Q=(K/4)1/2K1/2

Q=1/2K

K=2Q

TC = wL +rK

TC = 10(Q/2) +40(2Q)

TC=85Q

K=4L

L=K/4

L=Q/2

Input Prices and LR Cost Curves

- When the price of all inputs change by the same (percentage) amount, the optimal input combination does not change
- The same combination of inputs are purchased at higher prices

- C1=Isocost curve before ($200) and after ($220) a 10% increase in input prices

A

•

Q0

C1

0

L (labor

services/yr)

Example: A Shift in the Total Cost Curve When Input Prices Rise 10%

TC(Q) new

TC(Q) old

220

200

Q (units/yr)

Q0

8.1.2 Average and Marginal Cost Functions

- Definition: The long run average cost function is the long run total cost function divided by output, Q.
- That is, the LRAC function tells us the firm’s cost per unit of output…

Long Run Average and Marginal Cost Functions

- Definition: The long run marginal cost function is rate at which long run total cost changes with a change in output
- The (LR)MC curve is equal to the slope of the (LR)TC curve

and Marginal Costs

- When marginal cost is less than average cost, average cost is decreasingin quantity. That is, if MC(Q) < AC(Q), AC(Q) decreases in Q.
- When marginal cost is greater than average cost, average cost is increasingin quantity. That is, if MC(Q) > AC(Q), AC(Q) increases in Q.
- When marginal cost equals average cost, average cost does not change with quantity. That is, if MC(Q) = AC(Q), AC(Q) is flat with respect to Q.

8.2 Economies and Diseconomies of Scale

- If average cost decreases as output rises, all else equal, the cost function exhibits economies of scale.
- -large scale operations have an advantage
- If average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale.
- -small scale operations have an advantage

Economies and Diseconomies of Scale

- Why Economies of scale?
- -Increasing Returns to Scale for Inputs
- -Specialization of Labour
- -Indivisible Inputs (ie: one factory can produce up to 1000 units, so increasing output up to 1000 decreases average costs for the factory)

Economies and Diseconomies of Scale

- Why Diseconomies of scale?
- -Diminishing Returns from Inputs
- -Managerial Diseconomies
- -Growing in size requires a large expenditure on managers
- -ie: One genius cannot run more than 1 branch

Typical Economies of Scale

Minimum Efficient Scale – smallest

Quantity where LRAC curve reaches

Its min.

AC(Q)

Economies of scale

Diseconomies of scale

0

Q (units/yr)

Q*

Economies of Scale

- When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that AC(Q) decreases with Q, all else equal.
- When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that AC(Q) increases with Q, all else equal.

Economies of Scale

- When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.
- Production Function => Returns to Scale
- Costs => Economies of Scale

Example: Returns to Scale and Economies of Scale

CRS IRS DRS

Production Function Q = L Q = L2 Q = L1/2

Labor Demand L*=Q L*=Q1/2 L*=Q2

Total Cost Function TC=wQ wQ1/2 wQ2

Average Cost Function AC=w w/Q1/2 wQ

Economies of Scale none EOS DOS

Measuring Economies of Scale -

Output Elasticity of Total Cost

- Economies of Scale can be measured using output elasticity of total cost; how cost changes when output changes

Measuring Economies of Scale -

Output Elasticity of Total Cost

- Economies of Scale are also related to marginal cost and average cost

If TC,Q < 1, MC < AC, so AC must be decreasing in Q. Therefore, we have economies of scale.

- If TC,Q > 1, MC > AC, so AC must be increasing in Q. Therefore, we have diseconomies of scale.
- If TC,Q = 1, MC = AC, so AC is just flat with respect to Q.

- Let Cost=50+20Q2
- MC=40Q
- IF Q=1 or Q=2, determine economies of scale
- (Let Q be thousands of units)

- TC=50+20Q2
- MC=40Q
- AC=TC/Q=50/Q+20Q
- Initially: MC=40(1)=40
- AC=50/1+20(1)=70
- Elasticity=MC/AC=40/70 – Economies of Scale
- Finally: MC=40(2)=80
- AC=50/2+20(2)=65
- E=MC/AC=80/65 – Diseconomies of Scale

8.3 Short-Run Cost Curves

- In the short run, at least 1 input is fixed

(ie: (K=K*)

- Total fixed costs (TFC) are the costs associated with this fixed input (ie: rk)
- Total variable costs (TVC) are the costs associated with variable inputs (ie:wL)
- Short-run total costs are fixed costs plus variable costs: STC=TFC+TVC

Short Run Total Cost, Total Variable Cost and Total Fixed Cost

STC(Q, K*)

TVC(Q, K*)

rK*

TFC

rK*

Q (units/yr)

Short Run Costs

Example:

Minimize the cost to build 80 units if Q=2(KL)1/2 and K=25. If r=10 and w=20, classify costs.

Q=2(KL)1/2

80=2(25L)1/2

80=10(L)1/2

8=(L)1/2

64=L

Short Run Costs

Example:

K*=25, L=16. If r=10 and w=20, classify costs.

TFC=rK*=10(25)=250

TVC=wL=20(64)=1280

STC=TFC+TVC=1530

and Short Run Total Cost Functions

- The firm can minimize costs better in the long run because it is less constrained.
- Hence, the short run total cost curve lies above the long run total cost curve almost everywhere.

Only at point A is short run minimized as well as long run

TC2/r

Q1

Long Run Expansion path

TC1/r

TC0/r

C

•

Q0

Short Run

Expansion path

Q0

A

•

•

B

K*

0

L

TC0/w TC1/w TC2/w

8.3.2 Short Run Average and Marginal Cost Functions

- Definition: The short run average cost function is the short run total cost function divided by output, Q.
- That is, the SAC function tells us the firm’s cost per unit of output…

Short Run Average and Marginal Cost Functions

- Definition: The short run marginal cost function is rate at which short run total cost changes with a change in input
- The SMC curve is equal to the slope of the STC curve

Short Run Average and Marginal Cost Functions

- In the short run, 2 additional average costs exist: average variable costs (AVC) and average fixed costs (AFC)

- To make an omelet, one must crack a fixed number of eggs (E) and add a variable number of other ingredients (O). Total costs for 10 omelets were $50. Each omelet’s average variable costs were $1.50. If eggs cost 50 cents, how many eggs in each omelet?
- AC=AVC+AFC
- TC/Q=AVC+AFC
- 50/10=$1.50+AFC
- $3.50=AFC

- To make an omelet, one must crack a fixed number of eggs (E) and add a variable number of other ingredients (O). Total costs for 10 omelets were $50. Each omelet’s average variable costs were $1.50. If eggs cost 50 cents, how many eggs in each omelet?
- $3.50=AFC
- $3.50=PE (E/Q)
- $3.50=0.5 (E/Q)
- 7=E/Q
- There were 7 eggs in each omelet.

Average fixed cost is constantly decreasing, as fixed costs don’t rise with output.

AFC

0

Q (units per

year)

Average variable cost generally decreases then increases due to economies of scale.

AVC

AFC

0

Q (units per

year)

The Long Run Average Cost Function

as an Envelope Curve

In the long run, a firm can adjust its capital to a level that is then fixed in the short run.

The long run average cost curve (LRAC) therefore forms an “envelope” or boundary around the various short run average cost curves (SAC) corresponding to different capital levels.

The Long Run Average Cost Function

as an Envelope Curve

SAC(Q,K3)

SAC(Q,K1)

AC(Q)

SAC(Q,K2)

•

•

•

0

Q1 Q2 Q3

Q (units per

year)

The Long Run Average Cost Function

as an Envelope Curve

When a firm minimizes cost in the short run, given capital chosen in the long run,

-AC=SAC (Point A, next slide)

-MC=SMC (Point B, next slide)

-SAC is not at its min (in general) (Point C, next slide)

At the MES:

-AC=SAC=MC=SMC and SAC is at a minimum (two slides hence)

Often a firm produces more than one product, and often these products are related:

-Pepsi Cola makes Pepsi and Diet Pepsi

-HP makes Computers and Cameras

-Denny’s Serves Breakfast and Dinner

Often a firm benefits from economies of scope by producing goods that are related; they share common inputs (or good A is an input for good B). Efficiencies often exist in producing related products (ie: no shipping between plants).

If a firm can produce 2 products at a lower total cost than 2 firms each producing their own product:

TC(Q1,Q2)<TC(Q1,0)+TC(0,Q2)

That firm experiences economies of scope.

If the cities maintains local roads, it costs are $15 million a year. If a private firm covers park maintenance, it costs are $12 million a year. If the city does both, it costs $25 million a year.

TC(Q1,Q2)=$25 million

TC(Q1,0)+TC(0,Q2)=$15 million + $12 million

TC(Q1,0)+TC(0,Q2)=$27 million

TC(Q1,Q2)<TC(Q1,0)+TC(0,Q2)

Economies of scope exist.

Often with practice a firm “gets better” at producing a given output; it cuts costs by being able to produce the good faster and with fewer defects.

Ie: The first time you worked on elasticities, each question took you 10 minutes and 10% were wrong. By the end of the course you’ll be able to calculate elasticities in 4 minutes with only 5% error (for example).

Economies of experience are efficiencies (cost advantages) resulting from accumulated experience (learning-by-doing).

The experience curveshows the relationship between average variable cost and cumulative production volume.

-As more is produced (more experience is gained), average cost decreases.

vs. Economies of Scale

Economies of experience occur once, while economies of scale are ongoing.

A large producer benefiting from economies of scale will increase average costs by decreasing production.

A large producer benefiting from economies of experience may safely decrease production

Chapter 8 Key Concepts

- Long-Run Costs:
- TC=wL+rK (if labor and capital are the only inputs
- AC=TC/Q
- MC=∆TC/ ∆ Q
- Economies of scale summarize how average cost changes as Q increases
- Economies of scale = AC decreases as Q increases
- Diseconomies of scale = AC increases as Q increases

Chapter 8 Key Concepts

- Short-Run Costs
- TFC=All costs of the FIXED input
- TVC=All total costs of the VARIABLE input
- STC=TFC+TVC
- SAC=STC/Q
- SMC=∆STC/ ∆Q
- AFC=TFC/Q
- AVC=TVC/Q
- SAC=AFC+AVC

Chapter 8 Key Concepts

- If one firm has lower costs producing two goods than two firms producing the goods individually, that firm enjoys ECONOMIES OF SCOPE
- If AC decreases as cumulative output increases, a firm enjoys ECONOMIES OF EXPERIENCE
- This effect decreases over time
- Calculators are important in Econ 281

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