CONSUMER’ S SURPLUS AND COMPENSATING VARIATION

I t is possible to derive a measure of the b e ne fit an

i n d iv id u a l gains from the provision of a good, through an examination o f the i n d i v i d u a l ' s ordinary demand curve f o r th a t good: the

ordinary demand curve represents the functional r e la ti o n s h i p between the q u a nt it y of a good,X, the in d iv id u a l demands,and i t s p r ic e , p, given his level o f income, 3. The i n d i v i d u a l , whose ordinary demand curve f o r good X is i l l u s t r a t e d in Figure 1.1, when deciding i f i t is worthwhile to pay P f o r one u n i t of X, compares tha t price with P^, the price which he is w i l l i n g to pay. Because P^ is

greater than P , the i nd iv id ua l w i l l purchase the u n i t o f X and w i l l receive a surplus from consumption amounting to the difference

between the amount a c t u a l l y spent, P , and the amount which the X

i n d iv id u a l would have been w i l l inging to spend, P^ - tha t i s , the area PwcePx in Figure 1.1. I f X is p e r f e c t l y d i v i s i b l e , consumption w i l l expand to XD,where demand equals price,and the to t a l welfare

D

gain or b e n e f it enjoyed by the consumer, is the area to the l e f t o f the ordinary demand curve and above the p r i c e , dbP : t h i s is the

X

consumer's s u rp lu s . The welfare gain, or b e n e f it , associated with a decrease in price from P to P 1 is the difference between the

X X

consumer's surplus before the change, dbP , and the consumer's X

surplus a f t e r the change, daP ' : i . e . P baP ' .

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---1---

•d : O r d i n a r y

Figure 1 . 1 : The Ordinary Demand Curve and Consumer's Surplus.

The consumer's surplus measures welfare changes in r e la t i o n to the ove rall change in q u a nt it y demanded, and t h i s is made up of two components: the s u b s t i t u t i o n e f f e c t - the change in qu antity demanded r e s u l t i n g from the s u b s t i t u t i o n o f r e l a t i v e l y higher priced goods to the r e l a t i v e l y lower priced good when i t s price f a l l s ; and the income e f f e c t - the change in qu an ti ty demanded which results from the change in real income, or purchasing power, caused by the price change. The s u b s t i t u t i o n and income e f f e c t s can be i l l u s t r a t e d using ind if fe re nc e curve analysis,^ as in Figure 1.2, which

1 An analysis of the income and s u b s t i t u t i o n e f f e c t s in Marshallian geometry can be found in Hicks (1943).

d : O r di na r y

d : C ompens at ed

Figure 1 .2: S u b s t i t u t i o n and Income E f f e c t s , the Compensated Demand Curve and the Compensating V a r i a t i o n f o r a Pr ice F a l 1.

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i l l ust r at e s the individual who is faced with a choice between good X and a numeraire good Y. Consumption equilibrium i n i t i a l l y

occurs at point A where the individual maximises his u t i l i t y , U-| , given the constraint of the budget line ß 3 . When the price

P P

of X f al l s from P to P ' , the budget line faced by the individual X X

pivots around the point ß to become _ß___ß and the individual

P y P y P 7 2

is able to achieve a higher level of u t i l i t y , l^. The substitution effect of the price fall can be isolated from the income effect by reducing the individuals budget to ß1 so that he is returned to his original level of u t i l i t y , U-|, but at the new price ratio Py: Px' . In Figure 1.2, this "compensation" results in a shifting of the budget line, parallel to ß ß , so that i t becomes

Py Px'

tangential to the indifference curve at point C: the movement from A to C, along the indifference curve U-| is the substitution

effect. The income effect is therefore the movement across indifference curves from point C to point B. Now i t is clear that the substitution effect has no welfare significance, as i t results in a movement along an indifference curve with no change in the consumer's u t i l i t y ,

and that i t is the effective change in real income which affects the individual's well being. The consumer's surplus method of welfare measurement, because i t reflects both the substitution and income effects,will not always provide an accurate indication of a change

in welfare, with the more appropriate method being the measurement of the amount of numeraire needed to return the consumer to his

original level of u t i l i t y . In Figure 1.2, this compensating variation is ß -ß' , the vertical difference between the parallel budget

pL _

2 The two points A and B and their respective prices, Px and P ' enables the construction of the ordinary demand curvexfor the good X.

l i n e s , or the amount the i nd iv id ua l is w i l l i n g to pay to enjoy the decrease in the price o f X.

To compare the consumer's surplus and compensating v a r ia ti o n c o n ce pt s, it is useful to construct a Hicksian compensated demand curve - a demand r e la t i o n s h i p which holds u t i l i t y constant rather than maintaining a fi xe d budget, as is the case with the Marshallian demand fu n c ti on . By the i nd iv id ua l paying the compensating v a r ia ti o n in Figure 1.2, u t i l i t y is kept constant and the two points of

relevance to the compensated demand curve are A and C f o r the 3

respective prices P and P '. _{x x} The area to the l e f t o f the compensated demand curve d* between the prices Px and Px', P acP ' can be shown

X A

to be the same as the compensating v a r i a t i o n , ß - g1derived from

4 PY

the i nd iff er e nc e curve analysis. For the case o f a price f a l l ,

i t is clear th a t the compensating v a r i a t i o n is less than the consumer's surplus (P abP ' ) , however f o r the case o f a price rise, a s i m i l a r analysis would show th a t the compensating v a r i a t i o n , or the i n d i v i d u a l ' s

w il lin gne ss to accept compensation to ensure th a t the price ri s e does

3 The mathematical formulation of the de riv ati on o f the ordinary demand curve X.=X-(P,ß) is based on the problem o f maximizing u t i l i t y , subject to a budget c o n s t r a i n t , th a t i s :

Max U = U(X) s . t . Zp-jX. = ß

The compensated demand function X -* = X ^ * ( P ,U ) is based on the solution o f the dual to the u t i l icy maximization problem, the minimization o f expenditure, subject to a u t i l i t y c o n s tr a i n t:

MinE = Zp-X- s . t . U(X) =

V

A more complete o u t l i n e of these solutions and t h e i r associated f i r s t and second order assumptions can be found in Henderson and Quandt (1971), p. 23 and p. 25.

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not take place, is g r eat er than the consumer's surplus measure for the same price change.

Two measurement a l t e r n a t i v e s t herefore a r i s e from the consideration of the compensating var i at i on c r i t e r i o n of welfare change: the measurement of the area under the compensated demand , and the measurement of the i n d i v i d u a l ' s income, or numeraire,

equivalent of the change which is proposed. While these two measurements are equi val ent , both will d i f f e r from the consumer's

surplus measurement (the area under the ordinary demand curve), unless the

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income e l a s t i c i t y of demand for the good is zero.