Production with Two Variable Inputs:Returns to Scale

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Microeconomics ECO402

Lesson 17

Production with Two Variable Inputs

There is a relationship between production and productivity.

Long-run production K& L are variable.

Isoquants analyze and compare the different combinations of K & L and output

The Shape of Isoquants

Capital

per year

In the long run both

labor and capital are

variable and both

experience diminishing

returns.

Q3 = 90

Q2 = 75

Q1 = 55

Labor per year

Diminishing Marginal Rate of Substitution

Reading the Isoquant Model

1) Assume capital is 3 and labor increases from 0 to 1 to 2 to 3.

Notice output increases at a decreasing rate (55, 20, 15) illustrating diminishing

returns from labor in the short-run and long-run.

2) Assume labor is 3 and capital increases from 0 to 1 to 2 to 3.

Output also increases at a decreasing rate (55, 20, 15) due to diminishing returns

from capital.

Substituting Among Inputs

Managers want to determine what combination if inputs to use.

They must deal with the trade-off between inputs.

The slope of each isoquant gives the trade-off between two inputs while keeping output

constant.

The marginal rate of technical substitution equals:

MRTS = - Changein capital/Change in labor input

MRTS = - ΔK

(for a fixed level of Q)

ΔL

Microeconomics ECO402

Marginal Rate of Technical Substitution

Capital

per year

Isoquants are downward

sloping and convex

like indifference

curves.

Q3 =90

Q2 =75

Q1 =55

5 Labor per month

Observations:

1. Increasing labor in one unit increments from 1 to 5 results in a decreasing MRTS from

1 to 1/2.

2. Diminishing MRTS occurs because of diminishing returns and implies isoquants are

convex.

MRTS and Marginal Productivity

· The change in output from a change in labor equals:

(MP L )( Δ L)

· The change in output from a change in capital equals:

)( Δ K)

(MP

· If output is constant and labor is increased, then:

(MP L )( Δ L) + (MP K )( Δ K) = 0

(MP L )(MP K ) = - ( Δ K/ Δ L) = MRTS

Isoquants When Inputs are perfectly substitutable

Labor per month

Microeconomics ECO402

Perfect Substitute

Observations when inputs are perfectly substitutable:

1) The MRTS is constant at all points on the isoquant.

2) For a given output, any combination of inputs can be chosen (A, B, or C) to

generate the same level of output (e.g. toll booths & musical instruments).

Fixed-Proportions Production Function

Capital

per

month

Labor

per month

Fixed-Proportions Production Function

Observations when inputs must be in a fixed-proportion:

1) No substitution is possible. Each output requires a specific amount of each input

(e.g. labor and jackhammers).

2)To increase output requires more labor and capital (i.e. moving from A to B to C

which is technically efficient).

A Production Function for Wheat

Farmers must choose between a capital intensive or labor intensive technique of

production.

Microeconomics ECO402

Isoquant Describing the Production of Wheat

Capital

(Machine

Point A is more

hour per

capital-intensive, and

120

year)

B is more labor-intensive.

100

ĆK = -10

ĆL = 260

Output = 13,800 bushels

per year

Labor

250

500

760

1000

(hours per year)

Observations:

1) Operating at A:

L = 500 hours and K = 100 machine hours.

2) Operating at B

· Increase L to 760 and decrease K to 90 the MRTS < 1:

MRTS = -ΔK

= -(10 / 260) = 0.04

ΔL

3) MRTS < 1, therefore the cost of labor must be less than capital in order for the

farmer substitute labor for capital.

4) If labor is expensive, the farmer would use more capital (e.g. U.S.).

5) If labor is inexpensive, the farmer would use more labor (e.g. India).

Microeconomics ECO402

Returns to Scale

Measuring the relationship between the scale (size) of a firm and output

1. Increasing returns to scale: output more than doubles when all inputs are doubled

Larger output associated with lower cost (autos)

One firm is more efficient than many (utilities)

The isoquants get closer together

Increasing Returns:

Capital

The isoquants move closer

(machine

together

hours)

Labor (hours)

2. Constant returns to scale: output doubles when all inputs are doubled.

· Size does not affect productivity

· May have a large number of producers

· Isoquants are equidistant apart

Capital

(machine

hours)

Decreasing returns to scale: output less than doubles when all inputs are doubled

Constant Returns:

Isoquants are

20 equally spaced

Labor (hours)

Microeconomics ECO402

3. Decreasing returns to scale: output less than doubles when all inputs are doubled

· Decreasing efficiency with large size

· Reduction of entrepreneurial abilities

· Isoquants become farther apart

Capital

(machine

hours)

Decreasing Returns:

Isoquants get further

apart

Labor (hours)

Returns to Scale in the Carpet Industry

The carpet industry has grown from a small industry to a large industry with some very large

firms.

Question

Can the growth be explained by the presence of economies to scale?

The U.S. Carpet Industry

Carpet Shipments, 1996

(Millions of Dollars per Year)

1. Shaw Industries

$3,202

6. World Carpets

$475

2. Mohawk Industries

1,795

7. Burlington Industries

450

3. Beaulieu of America

1,006

8. Collins & Aikman

418

4. Interface Flooring

820

9. Masland Industries

380

5. Queen Carpet

775

10. Dixied Yarns

280

Are there economies of scale?

Costs (percent of cost)

· Capital -- 77%

· Labor -- 23%

Large Manufacturers

Increased in machinery & labor

Doubling inputs has more than doubled output

Economies of scale exist for large producers

Small Manufacturers

Small increases in scale have little or no impact on output

Proportional increases in inputs increase output proportionally

Microeconomics ECO402

Constant returns to scale for small producers

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