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The term ‘marginal utility’ refers to the change in total utility associated with a small (marginal) change in the consumption of one commodity (x1), the consumption of the other commodity remaining constant (at say, x̅2).
The rate of change of total utility of x1 associated with a small change in the amount of good 1(dx1) or MU1 is expressed as a ratio:
MU1 = du/dx1 = u(x1 + dx1, x2) – u(x1, x2)/dx1
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Alternatively changes in utility associated with a small change in consumption of x1 can be calculated by multiplying the change in consumption in x1 by its marginal utility:
Du = MU1 dx1
We can define marginal utility of x2 in the same way:
MU2 = du/dx2 = u(x1, dx2 + x2) – u(x1x2)/dx2
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While calculating MU2 we hold the consumption of x1 constant.
The change in utility associated with a change in consumption of x2 is:
du = MU2dx2
The magnitude of marginal utility depends on the magnitude of utility. This depends on how we measure utility. If utility is multiplied by 2, then marginal utility would be multiplied by the same number. The utility function remains the same even after scalar multiplication and continues to represent the same preferences.
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Marginal Utility and MRS:
We know that the MRS, the slope of the indifference curves at a given bundle of goods, is the rate at which a consumer is just willing to substitute a small amount of x1 for x1. Now we shall show how a utility function w(x1, x2) can be used to measure the MRS.
Since MRS is calculated by observing the movement of the consumer on the same indifference curve, on which total utility remains the same at all points we have;
MU1dx1 + MU2dx2 = du = 0
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This means that a change in the consumption of each good (dx1dx2) keeps utility constant when the consumers moves along the same indifference curve. If we solve for the slope of the indifference curves we have;
MRS = dx2/dx1 = MU1/MU2
The MRS is always negative due to the law of substitution: if the consumer gets more of x1, he has to get less of x2 in order to stay on the same indifference curve and enjoy the same level of satisfaction or utility. Of course, we can ignore the minus sign and take absolute value of the ratio of the two marginal utilities.
The MRS is calculated from the actual rate of exchange R = p1/p2 which indicates the marginal willingness to pay for x1 in terms for x2 and vice-versa. So it can be measured from a person’s actual (observed) behaviour in the market place.
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The magnitude of the marginal utility function depends on the choice of utility function, which is arbitrary. This means that the utility function, and, therefore, its derivative, viz. the marginal utility function are not uniquely determined.
Since the magnitude of the marginal utility function depends on the utility function that we use it does not depend on actual behaviour of the consumer alone. And the MRS is the ratio of the marginal utilities it is independent of the particular transformation of the utility function we choose to use.
Likewise since a monotonic transformation is just relabeling of the indifference curves, MRS is independent of the particular theory chosen to represent the preferences even though the marginal utilities are changed by monotonic transformations.
Thus if we multiply utility by 2 the MRS becomes:
MRS = – 2MU1/2MU2.
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