1 © 2006 by Nelson, a division of Thomson Canada Limited Production Theory LECTURE 4 ECON 340...

1 © 2006 by Nelson, a division of Thomson Canada Limited

Production Theory

LECTURE 4ECON 340

MANAGERIAL ECONOMICS

Christopher Michael

Trent UniversityDepartment of Economics

2 © 2006 by Nelson, a division of Thomson Canada Limited2006 Thomson Nelson

• The Production Function

• The Short-Run Production Function

• Optimal Use of the Variable Input

• Long-Run Production Functions

• Optimal Combination of Inputs

• Returns to Scale

Topics


Overview• Managers must decide not only what to produce

for the market, but also how to produce it in the most efficient or least cost manner.

• Economics offers widely accepted tools for judging whether the production choices are least cost.

• A production function relates the most that can be produced from a given set of inputs. » Production functions allow measures of the

marginal product of each input.


• A Production Function is the maximum quantity from any amounts of inputs or

• Q = f (L,K)• If L is labour and K is capital, one popular functional

form is known as the Cobb-Douglas Production Function

• Q = • L • K is a Cobb-Douglas Production Function

• The number of inputs is often large. But economists simplify by suggesting some, like materials or labour, are variable, whereas plant and equipment is fairly fixed in the short run.

The Production Function


• Short Run Production Functions:» MAX output, from any set of inputs

» Q = f ( X1, X2, X3, X4, X5 ... )

FIXED IN SR VARIABLE IN SR _

Q = f ( K, L) for two input case, where K as Fixed

• A Production Function has only one variable input, labour, is easily analyzed. The one variable input is labour, L.

The Short-Run Production Function


• Total Product = Q * L

• Average Product = Q / L

» output per labour

• Marginal Product =ΔQ/ΔL orQ/L

» output attributable to last unit of labour applied

• Similar to profit functions, the Peak of MP occurs before the Peak of average product

• When MP = AP, we are at the peak of the AP curve


Elasticities of Production• The production elasticity of labour,

» EL = MPL / APL = (Q/L) / (Q/L) = (Q/L)·(L/Q)» The production elasticity of capital has the identical in

form, except K appears in place of L. • When MPL > APL, then the labour elasticity, EL > 1.

» A 1 percent increase in labour will increase output by more than 1percent.

• When MPL < APL, then the labour elasticity, EL < 1. » A 1 percent increase in labour will increase output by less than 1

percent.

• When MPL = APL , then the labour elasticity, EL = 1» A 1 percent increase in labour will increase output by 1 percent.


Short-Run Production Function Numerical Example

L Q MP AP 0 0 --- --- 1 20 20 20 2 46 26 23 3 4 5

70 92 110

24 22 18

23.33 23 22

Marginal Product

L 1 2 3 4 5

Average Product

Labour Elasticity is greater then one,for labour use up through L = 3 units


• When MP > AP, then AP is RISING» IF YOUR MARGINAL GRADE IN THIS CLASS IS

HIGHER THAN YOUR GRADE POINT AVERAGE, THEN YOUR G.P.A. IS RISING

• When MP < AP, then AP is FALLING» IF YOUR MARGINAL BATTING AVERAGE IS

LESS THAN THAT OF THE TORONTO BLUE JAYS, YOUR ADDITION TO THE TEAM WOULD LOWER THE JAY’S TEAM BATTING AVERAGE

• When MP = AP, then AP is at its MAX» IF THE NEW HIRE IS JUST AS EFFICIENT AS

THE AVERAGE EMPLOYEE, THEN AVERAGE PRODUCTIVITY DOESN’T CHANGE


Law of Diminishing Returns

INCREASES IN ONE FACTOR OF PRODUCTION, HOLDING ONE OR OTHER FACTORS FIXED, AFTER SOME POINT, MARGINAL PRODUCT DIMINISHES.

A Short-Run Lawpoint of

diminishingreturns

Variable input

MP


Stages of Production – TP, AP and MP

Increasing Returns

Decreasing Returns

Negative Returns

TP Q

L

AP,MP

L

AP

MP

TP

L3L2L1

L1 L2 L3

Stage 1 Ep>1

Stage 2

0<Ep<1 Stage 3 Ep<0

• Stage 1: average product rising.

• Stage 2: average product declining (but marginal product positive).

• Stage 3: marginal product is negative, or total product is declining.

Pt of Marginal Returns

Ep=0Ep=1


Optimal Use of the Variable Input• HIRE, IF GET MORE

REVENUE THAN COST• HIRE if

TR/L > TC/L• HIRE if the marginal

revenue product > marginal factor cost: MRP L > MFC L

• AT OPTIMUM,

MRPL = W MFC

MRPL MPL • MRQ = W

optimal labour

MRPL

W W MFC

L

wage

•


MRPL is the Demand for Labour

• If Labour is MORE productive, demand for labour increases

• If Labour is LESS productive, demand for labour decreases

• Suppose an EARTHQUAKEEARTHQUAKE destroys capital

• MPL declines with less capital, wages and labour are HURT

D L

D’ L

S L

W

L’ L


• All inputs are variable» greatest output from any set of inputs

• Q = f (L, K) is two input example

• MP of capital and MP of labour are the derivatives of the production function» MPL = Q /L or Q /L

• MP of labour declines as more labour is applied. Also the MP of capital declines as more capital is applied.

Long-Run Production Functions


Isoquants & LR Production Functions

• In the LONG RUN, ALL factors are variable

• Q = f (L, K)• ISOQUANTS -- locus of input

combinations which produces the same output (A & B or on the same isoquant)» MAGNITUDE of SLOPE of

ISOQUANT is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution

» MRTS = (K1-K2)/(L1-L2) = <>K/<>L

» MRTS = MPL/MPK

ISOQUANT MAP

B

A

C

Q1

Q2

Q3

K

L


• The objective is to minimize cost for a given output

• ISOCOST lines are the combination of inputs for a given cost, C0

• C0 = CL·L + CK·K • K = C0/CK - (CL/CK)·L• Optimal where:

» MPL/MPK = CL/CK·» Rearranged, this becomes the

equimarginal criterion

Equimarginal Criterion: Produce where MPL/CL = MPK/CK

where marginal products per dollar are equal

Figure 4.9

Q(1)

D

L

K

at D, slope of isocost = slope of isoquant

C(1)

Optimal Combination of Inputs


Example – Isocost / Isoquant• Deep Creek Mining Co.

» Cost per worker is $50 per period CL» Mining Equipment can be leased at $0.2 per brake

per horse power.

» Recall C0 = CL·L + CK·K

» C = 50L + 0.2K, rearranging K = C/.2 – (250/.2)L» K = C/0.2 – 250L

K

L

K=C/0.2 – CL

Q= f(L,K)


Use of the Equimarginal Criterion

• Q: Is the following firm EFFICIENT?

• Suppose that:» MP L = 30

» MPK = 50

» CL = 10 (labour cost)

» CK = 25 (capital cost)

• Labour: 30/10 = 3

• Capital: 50/25 = 2

• A: No!

• A dollar spent on labour produces 3, and a dollar spent on capital produces 2.

USE RELATIVELY MORE LABOUR!

• If spend $1 less in capital, output falls 2 units, but rises 3 units when spent on labour

• Shift to more labour until the equimarginal condition holds.

• That is peak efficiency.


Allocative & Technical Efficiency

• Allocative Efficiency – asks if the firm is using the least cost combination of inputs» It satisfies: MPL/CL = MPK/CK

• Technical Efficiency – asks if the firm is maximizing potential output from a given set of inputs» When a firm produces at point T

rather than point D on a lower isoquant, they firm is NOT producing as much as is technically possible. Q(1)

D

Q(0)T


Scale Efficiency

• Scale Efficiency – asks if the firm is using the lowest possible minimum average cost for all production processes – defined as the ratio of this lowest cost to the potential average cost of production process chosen


Overall Production Efficiency

• Overall Production Efficiency – the product of allocative, technical, and scale efficiencies» (Overall Production Efficiency) =

(Allocative Efficiency) x (Technical Efficiency) x (Scale Efficiency)


Returns to Scale• A function is homogeneous of degree n

» if multiplying all inputs by (lambda) increases the dependent variable byn

» Q = f (L, K)» So, f ( L, ) = n • Q

• Constant Returns to Scale is homogeneous of degree 1. » 10% more all inputs leads to 10% more output.

• Cobb-Douglas Production Functions are

homogeneous of degree +


Cobb-Douglas Production Functions• Q = α • L • K

is a Cobb-Douglas Production Function

• IMPLIES:

» Can be CRS, DRS, or IRS

if + 1, then constant returns to scale

if + < 1, then decreasing returns to scale

if + > 1, then increasing returns to scale

• Coefficients are elasticities is the labour elasticity of output, often about 0.33

is the capital elasticity of output, often about 0.67

which are E L and EK

Most firms have some slight increasing returns to scale


ProblemSuppose: Q = 1.4 L 0.70 K 0.35

1. Is this function homogeneous?

2. Is the production function constant returns to scale?

3. What is the production elasticity of labour?

4. What is the production elasticity of capital?

5. What happens to Q, if L increases 3% and capital is cut 10%?


Answers

1. Yes. Increasing all inputs by , increases output by 1.05. It is homogeneous of degree 1.05.

2. No, it is not constant returns to scale. It is increasing Returns to Scale, since 1.05 > 1.

3. 0.70 is the production elasticity of labour

4. 0.35 is the production elasticity of capital

5. %Q = EL• %L+ EK • %K = 0.70(+3%) + 0.35(-10%) = 2.1% -3.5% = -1.4%

1 © 2006 by Nelson, a division of Thomson Canada Limited Production Theory LECTURE 4 ECON 340...

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