cost curves - exeterpeople.exeter.ac.uk/ckotsogi/bee2016/ch08-07-08.pdf · marginal cost curves 44...

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Cost CurvesCost Curves

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1. Introduction: HiSense2. Long Run Cost Functions

•Shifts •Long Run Average and Marginal Cost Functions•Economies of Scale•Deadweight Loss: "A Perfectly Competitive MarketWithout Intervention Maximizes Total Surplus"

3. Short Run Cost Functions

4. The Relationship Between Long Run and Short Run Cost Functions

33

Definition: The relates minimized total cost to output, Q, to the factor prices (w and r).

Where: L* and K* are the long run input demand functions

long run total cost function

TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r)

44

a. What is the long run total cost function for production function Q = 50L1/2K1/2?

L*(Q,w,r) = (Q/50)(r/w)1/2

K*(Q,w,r) = (Q/50)(w/r)1/2

55

TC(Q,w,r) =

w[(Q/50)(r/w)1/2]+r[(Q/50)(w/r)1/2]

= (Q/50)(wr)1/2 + (Q/50)(wr)1/2

= (Q/25)(wr)1/2

66

b. What is the graph of the total cost curve when w = 25 and r = 100?

TC(Q) = 2Q

2

77

Q (units peryear)

TC (£ per year)

TC(Q) = 2Q

£4M.

Example: A Total Cost Curve

88

Q (units peryear)

TC (£ per year)

TC(Q) = 2Q

1 M.

£2M.


99

Q (units peryear)

TC (£ per year)

TC(Q) = 2Q

1 M. 2 M.

£2M.

£4M.


1010

Definition: shows minimized total cost as output varies, holding input prices constant.

Graphically, what does the total cost curve look like if Q varies and w and r are fixed?

The long run total cost curve

1111

L (labor services per year)

K

0•L0

Q0TC = TC0

Example: Movement Along the Cost Curve

1212


K

0•

•L0 L1

K0

Q0

Q1

TC = TC1

TC = TC0


3

1313


K

0•

•L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0


1414

Q (units per year)


K

TC (£/yr)

0

0•

•L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0


1515

Q (units per year)


K

TC (£/yr)

0

0

LR Total Cost Curve

Q0

TC0 =wL0+rK0

••

L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0


1616

Q (units per year)


K

TC (£/yr)

0

0

LR Total Cost Curve

Q0 Q1

TC0 =wL0+rK0

••

L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0

TC1=wL1+rK1


1717

Graphically, how does the total cost curve shift if wages rise but the price of capital remains fixed?

1818

L

K

0

TC0/r

Example: A Change in Input Prices

4

1919

L

K

0-w0/r

TC0/r

TC1/r

-w1/r


2020

L

K

•

•

0

A

B

-w0/r

TC0/r

TC1/r

-w1/r


2121

L

K

Q0•

•

0

A

B

-w0/r

TC0/r

TC1/r

-w1/r


2222

Q (units/yr)

TC (£/yr)

TC(Q) post

Example: A Shift in the Total Cost Curve

2323

Q (units/yr)

TC (£/yr)

TC(Q) ante

TC(Q) post


2424

Q (units/yr)

TC (£/yr)

TC(Q) ante

TC(Q) post

TC0


5

2525

Q (units/yr)

TC (£/yr)

TC(Q) ante

TC(Q) post

Q0

TC1

TC0


2626

How does the total cost curve shift if all input prices rise (the same amount)? For example, suppose that all input prices double…

2727

L (labor services/yr)

K (capital services/yr)

0

•A

Example: All Input Prices Change

2828



0

-w/r


2929



0

•A

Q0

-w/r


3030



0

•A

Q0

-w/r

1

2


No change in input minimizing choices, but total cost shifts up by the change in input prices!

6

3131

Example:

TC(Q,w,r) = (wr)1/2Q/25

TC(Q, λw, λr) = (λw)1/2(λr)1/2Q/25 =λ(wr)1/2Q/25 = λTC(Q,w,r)

3232

Definition: 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…

AC(Q,w,r) = TC(Q,w,r)/Q

The long run average cost function

3333

Definition:measures the rate of change of total cost as output varies, holding constant input prices.

The long run marginal cost function

MC(Q,w,r) =

{TC(Q+ΔQ,w,r) – TC(Q,w,r)}/ΔQ

= ΔTC(Q,w,r)/ΔQ

where: w and r are constant

3434

Recall that, for the production function Q = 50L1/2K1/2, the total cost function was TC(Q,w,r) = (Q/25)(wr)1/2. If w = 25, and r = 100, TC(Q) = 2Q.

3535

a. What are the long run average and marginal cost functions for this production function?

AC(Q,w,r) = (wr)1/2/25

MC(Q,w,r) = (wr)1/2/25

b. What are the long run average and marginal cost curves when w = 25 and r = 100?

AC(Q) = 2Q/Q = 2.

MC(Q) = Δ(2Q)/ΔQ = 2.

3636

0

AC, MC (£ per unit)

Q (units/yr)

AC(Q) =MC(Q) = 2

£2

Example: Average and Marginal Cost Curves

7

3737

0


Q (units/yr)

AC(Q) =MC(Q) = 2

£2

1M


3838

0


Q (units/yr)

AC(Q) =MC(Q) = 2

£2

1M 2M


3939

Suppose that w and r are fixed…

•When marginal cost is less than average cost, average cost is decreasing in quantity. That is, if MC(Q) < AC(Q), AC(Q) decreases in Q.

4040

•When marginal cost is greater than average cost, average cost is increasing in 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.

4141

Q (units/yr)

AC, MC (£/yr)

0

AC

“typical” shape of AC


4242

Q (units/yr)

AC, MC (£/yr)

0

MC AC

“typical” shape of AC, MC

•


(both derived from TC!)

8

4343

Q (units/yr)

AC, MC (£/yr)

0

MC AC

AC at minimum when AC(Q)=MC(Q)

“typical” shape of AC, MC

•


4444

Definition: If average cost decreases as output rises, all else equal, the cost function exhibits economies of scale.

Similarly, if the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale.

Definition: The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.

4545

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.

4646

•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.

•When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.

4747

Definition: The percentage change in total cost per one percent change in output is the output elasticity of total cost, εTC,Q.

εTC,Q = (ΔTC/ΔQ)(Q/TC) =

= MC/AC

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

•If εTC,Q > 1, MC > AC, so AC must be increasing in Q. Therefore, we have diseconomies of scale.

•If εTC,Q = 1, MC = AC, so AC is just flat with respect to Q.

4848

Definition: tells us the minimized total cost of producing Q units of output, when (at least) one input is fixed at a particular level.

Definition: is the minimized sum of expenditures on variable inputs at the short run cost minimizing input combinations.

The total variable cost function

The short run total cost function

9

4949

STC(Q,K0) = TVC(Q,K0) + TFC(Q,K0)

Where: K0 is the fixed input and w and r are fixed (and suppressed as arguments)

Definition: The total fixed cost function is a constant equal to the cost of the fixed input(s).

5050

Q (units/yr)

TC (£/yr)

TFC

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

5151

Q (units/yr)

TC (£/yr)

TVC(Q, K0)

TFC


5252

Q (units/yr)

TC (£/yr)

TVC(Q, K0)

TFC

STC(Q, K0)


5353

The firm can minimize costs at least as well in the long run as in the short run because it is “less constrained”.

Hence, the short run total cost curve lies everywhere above the long run total cost curve.

5454

However, when the quantity is such that the amount of the fixed inputs just equals the optimal long run quantities of the inputs, the short run total cost curve and the long run total cost curve coincide.

10

5555

L

K

TC0/w

TC0/r

0

Example: Short Run and Long Run Total Costs

5656

L

K

TC0/w TC1/w

TC1/r

TC0/r

•

0

BK0


5757

L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r •••

0

A

C

B

Q1

K0


5858

L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r

Q0•

••

Expansion path

0

A

C

B

Q1

Q0

K0


5959

0

Total Cost (£/yr)

Q (units/yr)

TC(Q)

STC(Q,K0)

Q0

K0 is the LR cost-minimisingquantity of K for Q0

Q1


6060

0 Q (units/yr)

TC(Q)

STC(Q,K0)

•

Q0


Q1

ATC0Example: Short Run and Long Run Total Costs

Total Cost (£/yr)

11

6161


0 Q (units/yr)

TC(Q)

STC(Q,K0)

•

Q0


Q1

•A

C

TC0

TC1

Total Cost (£/yr)

6262


0 Q (units/yr)

TC(Q)

STC(Q,K0)

•

Q0


Q1

••

AC

B

TC0

TC1

TC2

Total Cost (£/yr)

6363

Definition: is the short run total cost function divided by output, Q.

That is, the SAC function tells us the firm’s short run cost per unit of output…

SAC(Q,K0) = STC(Q,K0)/Q

Where: w and r are held fixed

The Short run average cost function

6464

Definition:measures the rate of change of short run total cost as output varies, holding constant input prices and fixed inputs.

SMC(Q,K0)={STC(Q+ΔQ,K0)–STC(Q,K0)}/ΔQ

= ΔSTC(Q,K0)/ΔQ

where: w,r, and K0 are constant

The short run marginal cost function

6565

Note: When STC = TC, SMC = MC

STC = TVC + TFCSAC = AVC + AFC

Where:

SAC = STC/QAVC = TVC/Q (“average variable cost”)AFC = TFC/Q (“average fixed cost”)

6666

In other words,

The SAC function is the VERTICAL sum of the AVC and AFC functions

12

6767

Q (units peryear)

£ Per Unit

0

AFC

Example: Short Run Average Cost, AverageVariable Cost and Average Fixed Cost

6868

Q (units peryear)

£ Per Unit

0

AVC

AFC


6969

Q (units peryear)

£ Per Unit

0

SAC

AVC

AFC


7070

Q (units peryear)

£ Per Unit

0

SMCSAC

AVC

AFC


7171

Q (units peryear)

£ per unit

0

• ••

AC(Q)

SAC(Q,K3)

Q1 Q2 Q3

7272

Q (units peryear)

£ per unit

0

• ••

AC(Q)SAC(Q,K1)

Q1 Q2 Q3

13

7373

Q (units peryear)

£ per unit

0

• ••

AC(Q)SAC(Q,K1)

SAC(Q,K2)

Q1 Q2 Q3

7474

Q (units peryear)

£ per unit

0

• ••

AC(Q)SAC(Q,K1)

SAC(Q,K2)

SAC(Q,K3)

Q1 Q2 Q3

7575

Q (units peryear)

£ per unit

0

MC(Q)Example: Putting It All Together

7676

Q (units peryear)

£ per unit

0

AC(Q)

MC(Q)Example: Putting It All Together

7777

Q (units peryear)

£ per unit

0

••

AC(Q)SAC(Q,K2)

Q1 Q2 Q3

MC(Q)

SMC(Q,K1)

Example: Putting It All Together

7878

Q (units peryear)

£ per unit

0

• ••

AC(Q)SAC(Q,K1)

SAC(Q,K2)

SAC(Q,K3)

Q1 Q2 Q3

MC(Q)

SMC(Q,K1)


14

7979

Q (units peryear)

£ per unit

0

• ••

AC(Q)SAC(Q,K1)

SAC(Q,K2)

SAC(Q,K3)

Q1 Q2 Q3

MC(Q)

SMC(Q,K1)


8080

1. Long run total cost curves plot the minimized total cost of the firm as output varies.

2. Movements along the long run total cost curve occur as output changes. Shifts in the curve occur as input prices change.

3. Average costs tell us the firm’s cost per unit of output. Marginal costs tell us the rate of change in total cost as output varies.

4. Relatively high marginal costs pull up average costs, relatively low marginal costs pull average costs down.

cost curves - exeterpeople.exeter.ac.uk/ckotsogi/bee2016/ch08-07-08.pdf · marginal cost curves 44...

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