Topic 2 Lecture 13

Robin Naylor, Department of Economics, Warwick

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Topic 2 Lecture 13

• Isoquants, the Short-run production function, Marginal product of labour, and

firm’s costs.

Production isoquants and the MRTS

A household consumes x and y and derives Utility.

x and y are inputs and utility is an output.

We represent the relationship with indifference curves.

For a firm, K and L are inputs and X is the output.

We represent the relationship with production isoquants.

Also think about concept of a trade-off along

the Isoquant. B&B ch. 6 explores in detail the idea of the

production iso-quant (and the analogy with topographic

mappings with contour lines).

x

K

L

U

y

X

Slope of IC is MRS

U = U(X, Y)

Slope of Iso-quant is MRTS

X = X(K, L)


Topic 2: Lecture 13

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Think about a 3-D map of a mountain.

The mountain can be represented in a contour map (which is a view from above). The analogy is with an iso-quant diagram. The axes are K and L. We can consider how different output levels can be achieved by changing K and/or L.

The mountain can also be shown in profile (the view from one side). If we view the mountain from the south, we can see how height varies along its east-west profile from a particular southern viewpoint – but are unable to discern how height changes along the south-north dimension. The analogy is with the short-run production function, which shows how output varies with L for a particular level of K.

As my focus is on the short-run (in which K is constant) the production function is the diagram I am most interested in.


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Topic 2 Lecture 13

Sometimes we abbreviate this to :

( )X X L

In the short-run, K is fixed:

X

L

( , )X X K L

The short-run production function: X = X(L)

In the short-run the firm can vary only Labour inputs.

Labour is variable and so the costs of employing Labour are variable.

The amount of Capital, and hence the costs of employing Capital, are Fixed.


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Topic 2 Lecture 13

X

L


Assumption: The slope of the short-run production function is positive, but it is decreasing . . .

(Note – if you want to relate this to the iso-quant diagrams, see B&B page 233, Figure 6.19)


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Topic 2 Lecture 13

X

L


Slope of

X=X(L)

L

In terms of Economics, what is the slope of X(L)?


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Topic 2 Lecture 13

X

L


MPPL

L

dL

dX( )

is the change in output

when one extra unit of

L is employed.

Here we are assuming diminishing

MPPL (i.e., 'DRL').

X X L

dXMPPL

dL


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Topic 2 Lecture 13Digression on the relationship between MPPL and MRTS:

Consider the production function:

( , )

Totally differentiate:

(Interpret this in words)

Along the Iso-quant, dX = 0: thus,

X X K L

X XdX dL dK

L K

XdL

L

0, or,

=>

XdK

KX

X X dK MPPLLdK dL MRTSXK L dL MPPKK


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Topic 2 Lecture 13

X

L

X = X(L)

From the shape of the Short-run production function, we can infer the shape of the firm’s MPPL curve, as we have seen, and also:

(i) the shape of the firm’s Short-run Marginal Cost (SMC) curve and

(ii) the shape of the firm’s Short-run Total Variable Cost (STVC) curve.

(For now, we are considering only the firm’s Variable Costs.)

DRL


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Topic 2 Lecture 13

X

L

X = X(L)

From the shape of the Short-run production function, we can infer the shape of the firm’s MPPL curve and also:

(i) the shape of the firm’s Short-run Marginal Cost (SMC) curve

DRL


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Topic 2 Lecture 13

X

L

X = X(L)From the shape of the Short-run production function, we can infer the shape of the firm’s MPPL curve and also:


DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

What is the Marginal Cost of raising output by one unit from X1. . . . ?

= ?


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Topic 2 Lecture 13

X

L



DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

What is the Marginal Cost of raising output by one unit from X2 . . . . ?

= ?

dL1


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Topic 2 Lecture 13

X

L



DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

What is the Marginal Cost of raising output by one unit from X2 . . . . ?

= ?

dL1 dL2


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Topic 2 Lecture 13

X

L



DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

So:

SMC(X1) = w.dL1 and

SMC(X2) = w.dL2 =>Thus, SMC1<SMC2.

dL1 dL2


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Topic 2 Lecture 13

X

L



DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

So:

SMC(X1) = w.dL1 and


dL1 dL2

SMC1

SMC2


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Topic 2 Lecture 13

X

L



DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

So:

SMC(X1) = w.dL1 and


dL1 dL2

SMC1

SMC2

SMC


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Topic 2 Lecture 13

X

L

X = X(L)Three ways of showing the same thing . . .

. . . DRL

DRL

X

SMC

dX1

dX2

X1

X2

X1 X2

dL1 dL2

SMC1

SMC2

SMCMPPL

DRL DRL

MPPL


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Topic 2 Lecture 13

X

L

X = X(L)

From the shape of the Short-run production function, we can also infer :

(ii) the shape of the firm’s Short-run Total Variable Cost (STVC) curve.

DRL

X

SMC

SMC

X

STVC

?

DRL


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Topic 2 Lecture 13

X

L

X = X(L)

The STVC shows us what happens to the firm’s total Labour costs as output (and hence labour employment) increases.

STVC certainly increasing: but is it linear? Or is it getting steeper? Or flatter?

As SMC is rising under DRL, it follows that STVC is getting steeper: Why?

DRL

X

SMC

SMC

X

STVC

?

DRL

DRL


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Topic 2 Lecture 13

X

L

X = X(L)

As SMC is rising under DRL, it follows that STVC is getting steeper: Why?

Mathematically, what is the relationship between STVC and SMC?

DRL

X

SMC

SMC

X

STVC STVC

DRL

DRL

DRL


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Topic 2 Lecture 13

X

L

X = X(L)

DRL

X

SMC

SMC

X

STVC STVC

DRL

DRL

DRL

L

MPPL

MPPLDRL


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Topic 2 Lecture 13

X

L

X = X(L)

X

SMC

?

X

STVC

?

IRL

IRL

IRL

L

MPPL

?

IRL


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Topic 2 Lecture 13

X

L

X = X(L)

X

SMC

?

X

STVC

?

CRL

CRL

CRL

L

MPPL

?

CRL


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Topic 2 Lecture 13

X

L

X = X(L)

X

SMC

?

X

STVC

?

DRL

L

MPPL

?

IRL


Topic 2: Lecture 13

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Now read B&B 4th Ed., Chapter 6 on Inputs and Production Functions and Chapter 7 on Cost-minimisation and Chapter 8 on Cost Curves.

If you want to follow B&B in the same order as the material in lectures, you might start with section 6.3 on p. 210. Read as far as p. 219 and then go back to pp. 201-210. There is no need to study pp. 220-239 (but don’t let me stop you . . . These pages will deepen your understanding).

Chapter 7 goes into more detail than you need to follow the lecture material. You should focus on pp. 245-254, 270-271. In lectures, I avoid the need to use the idea of the iso-cost line – but Chapter 8 will be easier to follow if you make some effort to follow the idea of the iso-cost line in Chapter 7.

In Chapter 8, you should focus on pp. 292-310.

Topic 2 Lecture 13

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