© pilot publishing company ltd. 2005 chapter 8 production function & cost function

© Pilot Publishing Company Ltd. 2005

Chapter 8 Production Function

& Cost Function


• Short Run and Long Run• Production Function• Cost Accounting of Factor Inputs• Cost Function• Short Run Cost Curves• Long Run Cost Curves• Theory of Production Cost by A. Alchian• Advanced Material 8.1: Costs Related to an Owned Durable Equipment

Contents:


Short Run & Long Run


Reasons for the existence of different production runs: Whenever a market situation changes a firm has to make a new decision so as to maximize wealth.

The firm is uncertain if the change in the market situation is temporary or permanent.

It will make only the minimal and necessary change in factors to minimize cost.

At the beginning


Reasons for the existence of different production runs:

Even if the change is certain to be permanent,

the adjustment in factors should still be slow and gradual because hasty change involves a larger cost.

Afterwards


Reasons for the existence of different production runs:

Since adjustment is gradual, according to the completeness in the adjustment in factors, three different production runs are classified.


Classification of production runs

Very short run (VSR) all factors are fixed (remains unchanged).

Short run (SR) some factors are varied but some are fixed.

Long run (LR) all factors are variable and all required variations have been made.


Variable factors versus fixed factors

Variable factors: are factors of which the employment varies with output.

Fixed factors: are factors of which the employment does not vary with output.


Q8.1 Fill in the table.

VSR SR LR

Any change in factor

employment?

Any change in output level?

Reasons

No

No

Some factors are varied but some are fixed

Yes

All factors are variable and all required changes are made

YesNo adjustment —Time is needed to recognize the change, make decision & implement adjustment

Temporary adjust.—Time is needed to identify if the change is permanent & to make gradual adjustment to minimize cost

Final adjustment – Time is long enough for the final adjust. to be determined & implemented.


Q8.2:

“Some economists define the very long run as

the period over which the technology changes.”

Comment..


Production Function


Production function (生產函數 ) describes the relationship between inputs ( 投入量 ) and output (產量 ).

OutputInputs

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Variable

factor Variable

factor

Assumptions:

only two factors are involved capital & labour

Production function in the short run

CapitalCapital

Fixed Factor

Fixed Factor


____________________: is the whole amount of output produced by all the factors employed.

TP = Q

____________________: is the output per unit of the variable factor employed.

LQ

LTPAP

Three variables are defined to measure the output:

____________________: the change in output resulting from employing an additional unit of the variable factor.

11

LL L

Q

L

TPMP

Total product (TP)

Average product (AP)

Marginal product (MP)


Law of diminishing marginal productivity (邊際產量遞減定律 ) [or the law of diminishing returns (邊際回報遞減定律 ) or the law of variable proportions (可變比例定律 )]

The law of diminishing marginal productivity

states that if a _________ factor is added continuously to a

given amount of _________ factors, the marginal product

(and the average product) of the _________ factor must

finally decrease, ceteris paribus.

variable

fixed

variable


With a given amount of fixed factors, when one worker is employed, he can use only some of the fixed factors each time.

When more workers are employed, they can specialize and raise the productivity. (Both MP & AP ).

However, after all the fixed factors have been efficiently used, additional workers can help the preceding workers only. Hence MP which will finally drag down AP (and even TP).

Derivation of the law:


>

>>

Once the MP curve passes through the AP curve and lies below it, the AP curve will also be dragged down. Why?

Once the MP curve passes through the AP curve and lies below it, the AP curve will also be dragged down. Why?

MP

Fixed factorVariable factors>

>> MP

AP

AP

Graphical illustration:


When AP rises, MP must lie above AP.

When AP rises, MP must lie above AP.

How can MP lying above AP become lying below it? (If MP curve is continuous, it must pass through the maximum point of AP curve..))

How can MP lying above AP become lying below it? (If MP curve is continuous, it must pass through the maximum point of AP curve..))

When AP falls, MP must lie below AP.

When AP falls, MP must lie below AP.


Features:

The slope of TP curve is MP.The slope of TP curve is MP.

The slope of the line joining the origin and a point on TP is AP.

The slope of the line joining the origin and a point on TP is AP.

Notice the points where MP = maxi.; MP=AP & MP = 0.

Notice the points where MP = maxi.; MP=AP & MP = 0.


Implications of the law (if the law is violated)

By adding units of fertilizer or worker continuously to a given plot of land, no matter how small its size is, TP can be increased continuously. Enough food can be produced to feed all the people in the world.

An infinitesimal piece of land is adequate to supply the amount of food required. Hence the supply of land is no longer scarce. Land price would drop to zero.

MP of workers cultivating superior land does not fall and is always larger than MP of workers cultivating inferior land. No inferior land would be cultivated.


If the amount of land is fixed, TP cannot be increased significantly ( ∵MP & AP ) to feed all the people in the world.

Hence to raise production, more land is needed. However, as the supply of land is limited and scarce, under competition, land price must be positive.

Once MP of superior land falls below MP of inferior land, inferior land will also be cultivated.

Implications of the law (if the law holds)


Production Function in the long run

Returns to scale (規模回報 ) refers to the change in output when all factors are increased by the same proportion.


Production Function in the long run

Increasing returns to scale

There are three kinds of returns to scale:

Constant returns to scale

Decreasing returns to scale


Increasing returns to scale (規模回報遞增 )

when all factors are increased by the same proportion, the total product increases more than proportionately.


Constant returns to scale (規模回報不變 )

when the production scale increases, the total product increases proportionately.


when the production scale increases, the total product increases less than proportionately.

Decreasing return to scale (規模回報遞減 )


Assumptions (in the LR):Assumptions (in the LR):

At the end

the production function has decreasing returns to scale

At the end

the production function has decreasing returns to scale

At the beginning

the production function has increasing returns to scale

At the beginning

the production function has increasing returns to scale


Cost Accounting of

Factor Inputs


For factors hired or employed by a firm:

The costs are (the value of) the highest-valued alternative use of the money spent in hiring them.

They are called explicit costs (顯性成本 ), as they involve a transfer of money.


For factors owned by a firm:

The costs of using these factors are (the value of) the highest-valued alternative uses of the factors.

They are called implicit costs (隱含成本 ) or imputed costs (設算成本 ), as they do not involve a transfer of money.


Q8.4:What is the cost to a firm of using an owned property as an office for a year, if the premise can be leased at $20 000 a month or sold at $5 million, given a market interest rate of 4% per annum?


Classification of costs of different factor inputs

Sunk cost (historical cost)

The cost of a past act.

As past options are not available at present, sunk cost cannot be avoided now. Sunk cost is not a (present or future) cost.

Bygone is bygone. It should have no effect on any present or future decisions.


Fixed cost

The cost of employing fixed factors.

It does not change with output.

It is a present cost paid for the use of fixed factors and hence it affects the net receipt.

It has no effect on MC & no effect on the determination of the wealth-maximizing output level.


Variable cost The cost of employing variable factors.

It changes with output.

It is a present cost paid for the use of variable factors & hence it affects the net receipt.

It affects marginal cost & hence it affects the wealth-maximizing output.


Q8.6:A restaurant is making a short run decision for its production next month. Identify if the following costs are sunk costs (SC), fixed costs (FC) or variable costs (VC).(a) Rent of the restaurant under a 2-year contract ( )(b) Wage payments ( )(c) Expenditure on meat and vegetables ( )(d) Water charges ( )(e) Electricity charges ( )(f) Acquisition cost of machines ( )(g) Continuing possession cost of machines ( )(h) Operating cost of machines ( )


Cost Function


Cost function (成本函數 ) describes the relationship between output and cost.

Output

= ???


Short-run Cost Curves


Measure of costs

Output changes Cost changes

Total cost (TC)

Change in cost can be expressed in three ways:

Marginal cost (MC)

Average cost (AC)


Total cost

is the whole amount of payments to all factors used in producing a given amount of output (Q), composed of: Total fixed cost (TFC): is the whole amount of payments to fixed factors.

Total variable cost (TVC): is the whole amount of payments to variable factors.


TVC = w x L• total variable cost:

Formula:

TC = TFC +TVCTotal Cost:

Assume two factors only: Capital (fixed factor) and labour (variable factor) L units of labour are employed at a wage rate of w.

a constant independent of output• total fixed cost:


Average cost/average total cost (ATC)

is the cost per unit of output, composed of :

•average fixed cost (AFC): the fixed cost per unit of output.•average variable cost (AVC): the variable cost per unit of output.


Formula:

AVCAFCQ

TVCTFCQ

TCATC

Average Total Cost:

QTFCAFC average fixed cost:

AP

w

L

Qw

L

QL

Lw

Q

Lw

Q

TVCAVC

average variable cost:


ATC curve and AVC curve will come closer and closer as the amount of output increases (∵ATC = AFC + AVC and AFC drops continuously).

ATC curve and AVC curve will come closer and closer as the amount of output increases (∵ATC = AFC + AVC and AFC drops continuously).

AVC curve is U-shaped. (∵ AVC = w/AP and AP is inverted-U shaped.)

AVC curve is U-shaped. (∵ AVC = w/AP and AP is inverted-U shaped.)

AFC curve drops continuously. (∵AFC = TFC/Q)

AFC curve drops continuously. (∵AFC = TFC/Q)

Features:


The turning point of ATC curve (b) occurs at a larger output than the turning point of AVC curve (a). Why?

The turning point of ATC curve (b) occurs at a larger output than the turning point of AVC curve (a). Why?

(b)(a)

∵ At (a), the fall in AFC is > the rise in AVC initially

but at (b), the fall in AFC is < the rise in AVC eventually

∵ At (a), the fall in AFC is > the rise in AVC initially

but at (b), the fall in AFC is < the rise in AVC eventually


Marginal Cost

is the change in total cost for producing an additional unit of output, composed of :

• marginal fixed cost (MFC): is the change in fixed cost for producing an additional unit of output

• marginal variable cost (MVC): is the change in variable cost for producing an additional unit of output.


Formula:

0

Q

TFCMFC marginal fixed cost:

MVCMFCQ

TVCTFCQ

TCMC

Marginal cost:

marginal variable cost:

MP

w

L

Qw

L

QL

Lw

Q

Lw

Q

TVCMVC


MC curve passes through the minimum points of AVC curve and ATC curve.

MC curve passes through the minimum points of AVC curve and ATC curve.

MC or MVC curve is

U-shaped

As TFC is a constant, MFC = 0. So MC = MVC. As TFC is a constant, MFC = 0. So MC = MVC.

MC = MVC = w/MP. As MP curve is inverted-U shaped, MC or MVC curve is U-shaped.

MC = MVC = w/MP. As MP curve is inverted-U shaped, MC or MVC curve is U-shaped.


MC curve (= MVC curve) = Slope of TC curve & TVC curve.

MC curve (= MVC curve) = Slope of TC curve & TVC curve.

Derivation of total cost curves:

Notice the points where MC = mini.; MC = AVC and MC = ATC.

Notice the points where MC = mini.; MC = AVC and MC = ATC.


Q8.7: The following table is composed of product items and cost items of a firm. Suppose the unit cost of capital and labour are $10 and $20 respectively. Fill in the missing columns..

Units

of capital

Units

of labour

TP AP MP TFC TVC TC ATC

4

4

4

4

4

4

1

2

3

4

5

6

2

5

10

14

14

12


Q8.8 (a) When output increases, if AP of a variable factor rises, what will happen to AVC and ATC?

(b) When output increases, if AP of a variable factor falls, what will happen to AVC and ATC?


Long-run Cost Curves


>>

> >

>>

The firm enjoys economies of scale at the beginning

The firm enjoys economies of scale at the beginning

LRAC & LRMC LRAC & LRMC LRAC & LRMC LRAC & LRMC

>> As the scale of production further, the firm suffers disec

onomies of scale

As the scale of production further, the firm suffers disec

onomies of scale

LRAC & LRMCLRAC & LRMCLRAC & LRMCLRAC & LRMC


The production scale (combination of factors) with the lowest LRAC.

Optimum scale

LRAC curve with a horizontal region

U-shaped LRAC curve

Optimum scale


LRMC = slope of LRTCLRMC = slope of LRTC

Slope =LRMC=LRAC

Derivation of total cost curves:

Notice the points where LRMC = mini. and LRMC = LRAC.

Notice the points where LRMC = mini. and LRMC = LRAC.


Theory of Production Cost by A. Alchian


Volume = Rate x Time Duration

Rate effect: To produce the same volume of output, the faster the rate, the higher the average cost. Why?

Volume effect: At the same rate of production, the larger the volume, the lower the average cost. Why?

Proportionate increase in both the rate and volume: If both the rate and the volume are increased proportionately, the average cost may fall at the beginning but it must rise eventually. Why?


Advanced Material 8.1Cost related to an owned durable equipment

1. Present value and future value

PVPV = $X / (1+r)= $X / (1+r)ttPVPV = $X / (1+r)= $X / (1+r)tt

Computing the present value (PV) of a future sum ($X) is called discounting (折現 ).

Present value

FVFVt t = $Y (1+r)= $Y (1+r)ttFVFVt t = $Y (1+r)= $Y (1+r)tt

Computing the future value (FV) of a present sum ($Y) is called compounding (複算 ).

Future value


2. Classification of costs related to an owned durable equipment

Acquisition cost (取得成本 ) is the amount forgone in acquiring the ownership of an asset.

Continuing possession cost (繼續擁有成本 ) is the amount forgone in keeping an asset without using it.

Operating cost (操作成本 ) is the amount forgone in using an asset.


-Purchase price of machine A = $10 000-Immediate resale value = $9 600-Resale value if machine A has been laid idle for a year = $8 000-Resale value if machine A has been operated for a year = $4 000-Additional expense for operating machine A (e.g., labour cost) = $1 000-Market interest rate = 10%

(a) Acquisition cost = ?

(b) Continuing possession cost = ?

(c) Operating cost = ?

Worked example:


Correcting Misconceptions:

1. In the long run, all factors are variable and are varied gradually so as to minimize cost.

2. The law of diminishing returns states that when all inputs increase, output will increase at a decreasing rate eventually.

3. If the law of diminishing marginal productivity does not hold, scarcity no longer exists and production involves zero cost.


4. The cost of using a factor is equal to zero if no explicit payment is involved.

5. The cost of using an owned asset is equal to the purchase price of the asset.

6. Fixed cost is the same as sunk cost.

Correcting Misconceptions:

7. When output increases, if AP of the variable factor falls, the AC rises.


Survival Kit in ExamQuestion 8.1:In a recession, an employer can earn enough revenue to cover the rent, labour cost, water and electricity charges and the maintenance cost of machines. However, the remaining revenue cannot cover their repayment to bank loans for buying machines. Suppose the machines have zero resale value. Is it irrational for the employer (a) to continue the running of the firm, and (b) to buy the machines?

© pilot publishing company ltd. 2005 chapter 8 production function & cost function

Documents

pilot publishing company

fixed factors variable

production function

classification of production

short run vsr

short run sr

long run lr

cost final adjustment