Elasticity of Substitution

To begin with, Let’s do MRS. MRS or Marginal Rate of Substitution is

the number of units of Good Y to be added for every 1 unit of Good X removed. It usually appertains to an indifference curve of a consumer’s utility function .

Nevertheless, it also appertains to an isoquant which happens to be the level curve of a producer’s production function.

MRSXY = -∆Y/∆X

The Graphs

Indifference Curve Isoquant

Elasticity of Substitution Let F(x,y) = c , a function of two variables, be

the function of a level curve. Then the slope of the level curve turns out to be

y’ = d(y)/d(x) = -F1’(x,y)/F2’(x,y). (According to partial differentiation of implicit functions.) Therefore, -y’, represented by Ryx equals,

Ryx = F1’(x,y)/F2’(x,y) = MPX/MPY Or MUX/MUY

Where MP = Marginal Product and MU= Marginal Utility

Elasticity of Substitution For each value of Ryx in interval I, there

corresponds a certain point (x,y) on the level

curve F(x,y) = c, and thus a certain value of y/x(Ryx≈-∆y/∆x).

The fraction y/x is therefore, a function of Ryx.

When F(x,y) = c, the elasticity of substitution between y and x is

σyx = ElRyx(y/x)

IMPLICATION σyx = ElRyx(y/x) Thus, σyx is the elasticity of the fraction y/x with

respect to the marginal rate of substitution or simply, how easy it is to substitute x for y. Roughly speaking, it is the % change in the fraction y/x when we move along the level curve F(x,y) = c.

Technically speaking,

σyx = %∆(y/x)

%∆ MRSYX

Also, σyx = -F1’F2’(xF1’+yF2’)

xy[(F2’)2F11”-2F1’F2’F12”+(F1’)2F22”]

Elucidation The subscript yx in RYX and MRSXY means the

change in y due to change in x. Therefore, the notation yx.

MRSyx strictly decreases as we progress along the level curve. For Indifference Curve, it is due to the law of Diminishing Marginal Utility. This means that one is willing to forgo less and less of Good X for Good Y as one derives the maximum utility from it. Therefore, the Indifference Curve helps solve utility maximisation problems.

MRSXY = MPx/MPy = -d(y)/d(x)

A Stimulating Problem Compute σyx for the Cobb-Douglas Function

Q = ALαKβ

(This is an isoquant level curve, since it involves labour = L and capital = K. An isoquant curve helps with the problems of cost minimization.)

Before the solution, one should know that a Cobb-Douglas Equation always has

σyx = 1 It is known as the constant elasticity of substitution. General form of a Cobb-Douglas Expression is,

F(x,y) = Axayb

Solu

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to t

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Conclusion As we saw in the previous example, σkl = 1 Therefore, it is verified that a Cobb-

Douglas Function always has constant elasticity of substitution.

For functions in which the answer comes with a variable, that variable tends to 0.

For example, in σyx = 1/(1+ξ), ξ→0.

What do we learn Through this presentation, we’ve come

to know about elasticity of substitution and how the concepts of Maths act as a handy tool for its application.

Also, we’re reminded of a few rudimentary level concepts of Economics.

Srishti

419, 1st YearEco(H)