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If we rule out perverse demand (price-quantity) relationship, as is shown by the Giffen example, we can speak of the inverse demand function. Such a demand function treats price as a function of quantity, i.e., what p1 would have to be, at each level of demand of x1 in order for the consumer to choose that level of the commodity.
The two demand functions are not intrinsically different from each other. They are just two different ways of measuring the same inverse relationship between price and quantity. In Fig. 7.16 we present an inverse demand curve which graphically represents such a function.
The Cobb-Douglas Example:
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The Cobb-Douglas demand for x1 is expressed as αm/p1. The same price quantity relationship can be expressed as p1 = αm/x1.
Economic Interpretation:
We know that when the consumer consumes both x1 and x2 in positive amounts, he reaches equilibrium by equating the absolute value of MRS with the price ratio
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|MRS| = p1/p2
This simply means that at the optimal level of demand for x1, the following condition has to be satisfied:
p1 = p2|MRS|
Thus, at the original level of demand for x1, p1 is exactly proportional to the absolute value of the MRS between x1 and x2.
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Let us suppose x2 is money to be spent on other goods. So the MRS measures how much money the consumer would be willing to sacrifice to acquire a little more of x1. In this case the MRS measures the marginal willingness to pay. Here p1 = |MRS| since p2 = 1. So p1 itself is measuring the marginal willingness to pay.
At each quantity of x, the inverse demand function measures how much money the consumer is willing go give up for a little more of x1 or, alternatively stated, how much money the consumer was willing to sacrifice for the last unit purchased of x1. For a very small amount of x1 the two come down to the same thing.
The normal downward sloping inverse demand curve has a new meaning. For a very small quantity of x1 the consumer is willing to sacrifice a lot of money — that is, a large quantity of all other goods in order to acquire a small quantity of x1. As x1 becomes larger the consumer is willing to give up less money, on the margin, to acquire a little more of x1.
With an increase in the consumption of x1, the consumer is willing to sacrifice less money, on the margin, to acquire a little more of x1. Thus, with an increase in the consumption of x1 the consumer’s marginal willingness to pay, in the sense of the marginal willingness to sacrifice x2 for x1, falls.
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