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Home » Economics Homework Help » Microeconomics Help » Substitution Marginal Rate
Substitution Marginal Rate
The concept of marginal rate of substitution is an important tool of indifference curve analysis of demand. The rate at which the consumer is prepared to exchange goods X and Y is known as marginal rate of substitution. In our indifference schedule I above, which reproduced in table, process his high level of satisfaction remains the same. It follows that one unit gain in X fully compensates him for the loss of 4 units of Y. it means that at this stage he is prepared to exchange 4 units of Y for one unit of X. therefore, at this stage consumer’s marginal rate of substitution of X for Y whose loss can just be compensated by one unit gain in X. In other words, marginal rate of substitution represents the amount of Y which the consumer has to give up for the gain of one additional unit of X so that his level of satisfaction remains the same.

In table, when the consumer moves from combination B to combination C on his indifference schedule he forgoes 3 units of Y for additional one unit gain in X. hence the marginal rate of substitution of X for Y is 3. Likewise when the consumer moves from C to D, and then from D to E in his indifference schedule, the marginal rate of substitution of X and Y is 2 and 1 respectively.

How to measure marginal rate of substitution on an indifference curve? Consider where an indifference curve is shown. When the consumer moves on to point A to B on this indifference curve he gives up AS of and takes up SB of X and remains on the same indifference curve (or, in other words at the same level of satisfaction). It means that the loss of satisfaction caused by giving up AS equals the gain in satisfaction due to the increase in good X by SB. It follows that the consumer is prepared to exchange AS of SB increase in X. in other words rate of substitution of X for Y (MRSxy) is equal to  AS/SB                                                                                             
Now a small change in the amount Y such as AS along an indifference curve can be written as ΔY and the change in the amount of X as ΔX. Thus, ΔY shows the amount which the consumer has to give up for the ΔX increase in X if he is to remain on the same indifference curve. Therefore, it follows that:

Marginal rate of substitution of X for Y (MRSxy) = AS/SB = ΔY/ΔX

Now suppose that points A and B are very close to each other so that it can be assumed that both of them lie on the same tangent t T. Now in a right angled triangle ASB. It therefore follows that: (MRSxy) = AS/SB = ΔY/ΔX = tangent of <ABS = tTO

Hence MRSxy = tangent of <tTO

But the tangent of <tTO is equal Ot/OT
                                                 
The tangent of <tTO indicates the slope of the tangent line tT drawn at point A or B on the  indifference curve on A or B is equal to the tangent of curve . In other words the slope of the indifference curve at appoint A or B is equal to the tangent of <tTO. It therefore follows:

MRSxy = tangent of <tTO = slope of the indifference curve on A or B = Ot/OT
                                                                                                      
It is clear from above that I few have to find out the MRSxy at a point on the indifference curve we can do so by drawing tangent at the point on the indifference curve and then measuring the slope by estimating the value of the tangent of the angle which the tangent line makes with the X-axis.
                                               
Combination Good X
Good Y MRSxy
A 1 12 4
B 2 8 -
C 3 5 3
D 4 3 2
E 5 2 1

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