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# Appendix 5: Origins of the Hicksian measures of consumer surplus

DOI link for Appendix 5: Origins of the Hicksian measures of consumer surplus

Appendix 5: Origins of the Hicksian measures of consumer surplus book

# Appendix 5: Origins of the Hicksian measures of consumer surplus

DOI link for Appendix 5: Origins of the Hicksian measures of consumer surplus

Appendix 5: Origins of the Hicksian measures of consumer surplus book

## ABSTRACT

Ambiguity arises simply because we are tomeasure the real gain in terms ofmoney income, as defined, along the vertical axis. Hicks’s compensating variation, CV, is equal to Y0Y1, for if the consumer is made to pay this sum in order to be permitted to buy x at the price (given by the slope of Y0X0), he could just reach Q0 on his original indifference curve I0. That is, if he is to be exactly as well off as he was originally before x was introduced, Y0Y1 is the maximum sum he can afford to pay for the privilege of buying x at the given price. And if called upon to pay this maximum, the amount of x that he would in fact buy is given here by OM0. Turning to Henderson’s distinction, we now ask a different question: what is

the minimum sum the consumer will accept to give up entirely the opportunity of buying the new good x at the market price, given by the slope of Y0X0? The

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income OY0, his 2 his indifference curve I1. He is then just as well off as he would have been if, at his original income OY0, he was able to buy x at the given price. This measure of consumer’s surplus was called by Hicks (1944) the equivalent variation, EV, inasmuch as, in the absence of x, such a sum provides the consumer with an equivalent improvement in his welfare. Provided the income effect is positive (‘normal’), Q1 will be to the right of Q0

on the parallel budget lines. OM1 will therefore be larger thanOM0, and Y0Y2 will be larger than Y0Y1.2

3 More generally, the definition of CV is the sum of money to be paid by the consumer when the price falls; or to be received by him when the price rises – which, following a change in the price, leaves him at his initial level of welfare. The EV, conversely, is that sum of money to be received by the consumer when the price falls; or to be paid by himwhen it rises – which, if he were exempted from the change in price, would yet provide him with the same welfare change. These two measures have already been illustrated in Figure A5.1 for the special case of the introduction of a newgood x at a given price, rather than for a change in the existing price of x. They are now illustrated inFigureA5.2 for the case of a fall in the existing price of x. Initial money income is again represented by OY0, initial real income being

given by the indifference curve I1 which is reached by the tangency of the price line p1 at Q1. If the price of x falls to p2, the tangency of the p2 price line at Q2 raises the consumer’s real income from I1 to I2. His CV is then equal to Y0Y1, this being the maximum sum he could afford to pay for the lower price p2 without being any worse off. For if he pays this sum, so reducing his money income to OY1, the now lower price p2 enables him to reach B on his original I1 curve. His EV, conversely, is equal to Y0Y2, this being theminimal sum hewill accept to forgo the opportunity to buy what he wants at the lower price p2, for with this sum, his total income would be equal to OY2, and with this income and the old price p1 he could just reach the higher indifference curve I2 at C. This increase in his welfare is exactly equal to that which he could attain with this new price p2 and with his original money income OY0. Once more, Y0Y2 will exceed Y0Y1 for a ‘normal’ good x, as drawn, the reverse being true if x, instead, were an ‘inferior’ good. We can nowgo through the same exercise for a rise in the price of x.With income

OY0, we begin with the consumer being faced with p2 and, therefore, choosing point Q2 on indifference curve I2. A rise in the price of x to p1 now induces his to take up the position Q1 on the I1 indifference curve. Our definitions would therefore measure the CV of such a price rise by Y2Y0, this being the minimum

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sum that would restore the individual’s welfare to its original level I2 when the price rises to p1. The EV is now to be measured as equal to Y0Y1, this being the maximum sum the consumer is prepared to give up if he is exempted from the higher price p1. For giving up this sum and retaining the old price p2 would enable him to reach It (at B), which is the level of welfare he reaches if he is not exempted from the rise in price to p1.