ECON 377/477

Topic 1.2

Production functions

Outline

• Introduction• Properties• Quantities of interest• An example• Short-run production functions• Transformation functions

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Introduction

• Consider a firm that uses amounts of N inputs to produce a single output

• The technological possibilities of such a firm can be summarised using the production function:

q = f(x)

where q is output and x = (x1, x2, …, xN)׳ is an Nx1 vector of inputs

• We assume these inputs are under the control of the decision maker

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Properties

• Non-negativity: the value of f(x) is a finite, non-negative, real number

• Weak essentiality: the production of positive output is impossible without the use of at least one input

• Non-decreasing in x (monotonicity): additional units of an input will not decrease output , i.e. if x0 ≥ x1, then f(x0) ≥ f(x1)

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Properties

• Concave in x: Any linear combination of the vectors x0 and x1 will produce an output that is no less than the same linear combination of f(x0) and f(x1)

Formally, f(θx0) + (1 – θ)x1 ≥ θ f(x0) + (1 – θ)f(x1)

If the production function is continuously differentiable, concavity implies that all marginal products (MPs) are non-increasing (the law of diminishing marginal productivity)

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Properties

• The diagram on the next slide depicts a production function defined over a single input, x

• The values of q are all non-negative and finite real numbers for the values of x represented on the horizontal axis (non-negativity)

• The function passes through the origin (weak essentiality)

• The MP of x is positive at all points between the origin and point G (monotonicity) but monotonicity is violated on the curved segment GR

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AP at E is the slope of the ray through the origin and E

q = f(x)

MP at G = 0

Point of optimal scale

E

Concavity violated0

Feasible region Monotonicity violated

G

D

q

x

R

Properties

• As we move along the production function from the origin to point D, MPx increases

• Thus, the concavity property is violated at these points

• But concavity is satisfied at all points on the curve segment DR

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Properties

• Extending the graphical analysis to the multiple-input case is difficult

• In such cases it is common practice to plot the relationship between two of the variables while holding all others fixed

• We now consider a two-input production function and plot the relationship between the inputs x1 and x2 while holding output fixed at the values q0

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Properties

• We also plot the relationship between the two inputs when output is fixed at the values q1 and q2, where q2 > q1 > q0 (output isoquants), shown in the diagram on the next slide

• If properties are satisfied, these isoquants are non-intersecting functions that are convex to the origin, as depicted

• The slope of the isoquant is the marginal rate of technical substitution (MRTS) that measures the rate at which x1 must be substituted for x2 in order to keep output at its fixed level

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F

MRTS at F = slope of the

isoquant at F

x2

x10

F(x1,x2) = q0

F(x1,x2) = q1

F(x1,x2) = q2

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Properties

• An alternative representation of a two-input production function is provided in the diagram on the next slide

• The lowest of the four functions plots the relationship between q and x1:

• The value of x2 is held fixed

• The other functions plot the relationship between q and x1 when x2 is fixed at different values

01 2 2( )q f x x x

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q

x10

41 2 2( )q f x x x

31 2 2( )q f x x x

21 2 2( )q f x x x

01 2 2( )q f x x x

11 2 2( )q f x x x

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Quantities of interest

• If the production function is twice-continuously differentiable we can use calculus to define a number of economic quantities of interest

• For example, two quantities we have already encountered are the MP and the MRTS

• Related concepts that do not depend on units of measurement are the output elasticity and the direct elasticity of substitution, which is usually denoted as σ in the two-input case

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Quantities of interest

• In the next slide, an infinitesimal movement from one side of point A to the other results in an infinitesimal change in the input ratio but an infinitely large change in the MRTS, implying that σ = 0

• Thus, in the case of a right-angled isoquant, an efficient firm must use its inputs in fixed proportions

• That is, no input substitution is possible

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A

x2

x10

σ = 0

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Quantities of interest

• In the next slide, a movement from D to E results in a large percentage change in the input ratio but leaves the MRTS unchanged

• This result implies that the isoquant is a straight line and inputs are perfect substitutes

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E

D

x2

x10

σ = infinity

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Quantities of interest

• In the next slide, an intermediate (and more common) case is depicted where σ lies somewhere between zero and infinity

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x2

x10

σ lies between zero and infinity

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Quantities of interest

• In the multiple-input case it is possible to define at least two other elasticities of substitution: the Allen partial elasticity of substitution (AES) and the Morishima elasticity of substitution (MES)

• The DES is sometimes regarded as a short-run elasticity because it measures substitutability between xn and xm while holding all other inputs fixed

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Quantities of interest

• Economists use the term, short-run, to refer to time horizons so short that at least one input is fixed

• The AES and MES are long-run elasticities because they allow all inputs to vary

• When there are only two inputs, DES = AES

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Quantities of interest

• The MP measures the output response when one input is varied and all other inputs are held fixed

• But we are often interested in measuring output response when all inputs are varied simultaneously

• If a proportionate increase in all inputs results in a less than proportionate increase in output, then we say the production function exhibits decreasing returns to scale (DRS)

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Quantities of interest

• If a proportionate increase in inputs results in the same proportionate increase in output, the production function is said to exhibit constant returns to scale (CRS)

• If a proportionate increase inputs leads to a more than proportionate increase in output the production function exhibits increasing returns to scale (IRS)

• There are many reasons why firms may experience different returns to scale

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Quantities of interest

• A widely used measure of returns to scale is the elasticity of scale or total elasticity of production

• The production function exhibits locally DRS, CRS or IRS as the elasticity of scale is less than, equal to or greater than 1

• For the Cobb-Douglas production function defined over N inputs:

q = ax1β1x2

β2 … xNβN

• The output elasticities are En = Σβn for n = 1, …, N

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An example

• Refer to pages 18-19 for an example illustrating the computation of MPs and elasticities in a two-input Cobb-Douglas production function

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Short-run production functions

• Short-run production functions are obtained by holding one or more inputs fixed

• Consider equation (2.10) on page 18 of CROB:

q = 2x10.5x2

0.4

• If x2 were fixed at 100 in the short run, the short-run production function would be:

q = 2x10.51000.4, = 12.619x1

0.5

• This function is depicted in the diagram on the next slide, along with another function based on the assumption that x2 is fixed at 150, when:

q = 14.841x10.5

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q = 2x10.51000.4, = 12.619x1

0.5

q = 2x10.51500.4, = 14.841x1

0.5

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0

5

10

15

20

25

30

35

0 1 2 3 4

Short-run production functions

Short-run production functions

• A family of short-run production functions could be constructed in this way, each of which satisfies the four properties outlined above

• As a group, this family could be viewed as a long-run production function because it depicts the production possibilities of the firm when both inputs vary

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Transformation functions

• The production function concept can be generalised to more than one output

• The technological possibilities of a firm that uses N inputs to produce M outputs can be summarised by the transformation function:

T(x,q) = 0

where q = (q1, q2, …, qM)׳ is an Mx1 vector of outputs

• Transformation functions are special cases of distance functions, which are discussed in detail later

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