Production and Cost Functions and Their Estimation

Production and Cost Functions and Their

Estimation

Production function

A table, graph, or equation showing the maximum output rate of the product that can be achieved from any specified set of usage rates of inputs

Production functionThomas Machine Company

Amount of Labor Output of Parts AP Labor MP Labor(annual # units) (hundreds/year)

1 12 12.0 12 27 13.5 153 42 14.0 154 56 14.0 145 68 13.6 126 76 12.7 87 76 10.9 08 74 9.3 -2

Production functionThomas Machine Company

0

20

40

60

80

0 2 4 6 8 10

Labor

Par

ts

Production functionThomas Machine Company

-5

0

5

10

15

20

0 5 10

Labor

Par

ts AP Labor

MP Labor

Law of diminishing marginal returns

If equal increments of an input are added to a production process, and the quantities of other inputs are held constant, eventually the marginal product of the input will diminish

Note: 1) This is an empirical generalization.

2) Technology remains fixed. 3) The quantity of at least one

input is held fixed.

Marginal revenue product

The amount that an additional unit of the variable input adds to the firm’s total revenue

MRPY = TR/Y

Marginal expenditure

The amount that an additional unit of the variable input adds to the firm’s total costs.

MEY = TC/Y

Optimal level of input use

MRPY = MEY

Production functions with two variable inputs

Number of Machine ToolsAmount of Labor 3 4 5 6

1 5 11 18 242 14 30 50 723 22 60 80 994 30 81 115 1255 35 84 140 144

1 2 3 4 5

Number ofMachine Tools

0

50

100

150

Labor

Q = f (labor, machine Tools)

Isoquant

A curve showing all possible (efficient) combinations of inputs that are capable of producing a certain quantity of output

Iso quant

same quantity

Labor

Capital

0

K2

100

200

300K1

L2 L1

Marginal rate of technical substitution

Shows the rate at which one input can be substituted for another input, if output remains constant. (Slope of the isoquant.)

Given Q = f(X1, X2)

MRTS = -X2 / X1

= -MP1 / MP2

Isocost curves

Various combinations of inputs that a firm can buy with the same level of expenditure

PLL + PKK = M

where M is a given money outlay.

Labor

Capital

0

M/PK

M/PL

Slope = -PK /PL

Maximization of output for given cost

Labor

Capital

0100

200300

R

MPL/PL = MPK/PK

Labor

Capital

0100

200300

R

Optimal Lot Size

• To consider the size of inventory

• Find the relationship between size of lot and total annual cost.

What Toyota Taught the World?

• Lower the cost per setup

• Reduce the optimal lot size

• Just-in-time production system

Returns to scale

If the firm increases the amount of all inputs by the same proportion:

• Increasing returns means that output increases by a larger proportion

• Decreasing returns means that output increases by a smaller proportion

• Constant returns means that output increases by the same proportion

Output elasticity

The percentage change in output resulting from 1 percent increase in all inputs.

> 1 ==> increasing returns < 1 ==> decreasing returns = 1 ==> constant returns

Example: Xerox

Sending out teams of engineers and technicians to visit other firms to obtain information concerning best-practice methods and procedures.

• Competitive Benchmarking

Measurement of Production Functions

Three types of statistical analysis •Time series data•Cross section data•Technical information

The Analysis of Costs

Opportunity costs

The value of the other products that the resources used in production could have produced at their next best alternative

Historical costs

The amount the firm actually paid for a particular input

Explicit vs. implicit costs

• Explicit costs include the ordinary items that an accountant would include as the firms expenses

• Implicit costs include opportunity costs of resources owned and used by the firm’s owner

Short run

A period of time so short that the firm cannot alter the quantity of some of its inputs

• Typically plant and equipment are fixed inputs in the short run

• Fixed inputs determine the scale of the firm’s operation

Three concepts of total costs

•Total fixed costs = FC•Total variable costs = VC

•Total costs = FC + VC

OUTPUT FC VC TC0 2000 0 20001 2000 100 21002 2000 180 21803 2000 280 22804 2000 392 23925 2000 510 25106 2000 650 26507 2000 800 28008 2000 960 29609 2000 1140 3140

10 2000 1340 334011 2000 1560 356012 2000 2160 4160

Fixed, variable, and total costs Media Corp.

Fixed, Variable, and Total Costs -- Media Corp.

010002000300040005000

0 10 20

Units of Output

dolla

rs

FC

VC

TC

Average and marginal costsMedia Corp.

OUTPUT AFC AVC ATC MC01 2000.0 100.0 2100.0 1002 1000.0 90.0 1090.0 803 666.7 93.3 760.0 1004 500.0 98.0 598.0 1125 400.0 102.0 502.0 1186 333.3 108.3 441.7 1407 285.7 114.3 400.0 1508 250.0 120.0 370.0 1609 222.2 126.7 348.9 180

10 200.0 134.0 334.0 20011 181.8 141.8 323.6 22012 166.7 180.0 346.7 600

Average and marginal costsMedia Corp.

0500

100015002000

0 2 4 6 8 10 12

Units of output

$$$ AFC

AVCATCMC

Long-run cost functions• Often considered to be the firm’s planning horizon

• Describes alternative scales of operation when all inputs are variable

Quantity of output

Average cost

Long-run average cost function

Shows the minimum cost per unit of producing each output level when any scale of operation is available

Quantity of output

Average cost

SR average cost functions

LR average cost

Key steps:Cost estimation process

Definition of costs Correction for price level changes

Relating cost to output Matching time periods Controlling product, technology, and plant

Length of period and sample size

Minimum efficient scale

The smallest output at which long-run average cost is a minimum.

Quantity of output

Average cost

Qmes

The survivor technique

• Classify the firms in an industry by size and compute the percentage of industry output coming from each size class at various times

• If the share of one class diminishes over time, it is assumed to be inefficient

• These firms are then operating below minimum efficient scale

Economies of scope

Exist when the cost of producing two (or more) products jointly is less than the cost of producing each one alone.

S = C(Q1) + C(Q2) - C(Q1+ Q2)

C(Q1+ Q2)

Break-even analysis

Quantity of output

Dollars

Total Revenue

Total Cost

Loss

Profit