1 This chapter is a simple exercise inpartial equilibrium analysis: inthe adding and subtracting of consumer surpluses arising from sequential or simultaneous changes in the prices of two or more related goods.
2 Hicks (1956) has shownhow the consumer surplus ontwo or more substitute goods, say gas and electricity, that are introduced simultaneously can be measured.
Suppose that gas is introduced at a given price pginto an area that has no electricity.
The shaded triangle of Figure 5.1 canbe takenas a measure of the resulting consumer surplus. If, following this event, electricity is introduced at a price pe, the demand curve for electricity DeEe, is obviously smaller whengas is available at a fixed price pgthanit would be inthe absence of gas, for the consumers already derive much benefit from gas, and the introduction of a fairly close substitute is not so great a boonas it would be if, instead, there had beenno gas inthe first place.
Dg
pg
O Og Eg g
Figure 5.1
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 33 — #2 Consumer surplus when prices change 33 The additional gain to consumers from introducing electricity into a gas-using area is givenby the shaded triangle inFigure 5.2. The sum of these two triangles together measures the consumer surplus from providing both gas and electricity at prices pgand pe, respectively.
Should the economist elect to measure the simultaneous introduction of gas and electricity using the same sequential device but in the reverse order (that is, first measuring the consumer surplus for introducing electricity when gas is assumed to be unavailable, and then measuring the consumer surplus for gas on the assumption that electricity is already available), the sum of these two component surpluses should, theoretically, be exactly the same.
This method of adding consumer surpluses can, of course, be extended to three or more goods, and is just as valid if the goods inquestionare complements rather than substitutes. If, for example, gas and electricity were complements – as they would be if the only use of electricity were the heating of electric pokers for lighting gas fires – a fall inthe price of gas would raise the demand curve for electricity.
The analysis is, of course, symmetrical for simultaneous withdrawal of two or more goods. Thus, assuming again that gas and electricity are close substi-tutes, if electricity is first withdrawnfrom the market while gas remains as readily available at its old price pgthe loss of consumer surplus is given by the shaded triangle in Figure 5.2. Since gas is a substitute, the demand curve for gas shifts to the right following the withdrawal of electricity. The resulting or final consumer surplus for gas thenbecomes the shaded triangle inFigure 5.1, and is the measure of the loss sustained if gas, previously available at pg, is also withdrawnfrom the market.
De
pe
O Oe Ee e
Figure 5.2
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 34 — #3 3 The further extension of the method to simultaneous price changes poses no problems. Suppose once more that electricity and gas are close substitutes and that the prices of both rise. The loss of consumer surplus arising from the rise inthe price of gas from pg1 to pg2, the price of electricity being (provisionally) unchanged, is shown by the shaded strip in Figure 5.3. As a direct result of the rise in the price of gas, the demand curve for electricity now moves outward from DeEeto DeEein Figure 5.4. If, following this adjustment, the price of electricity rises from pe1 to pe2, the further loss of consumer surplus is given by the shaded area in Figure 5.4. It is hardly surprising, after all, that the loss of consumer surplus from a rise inthe price of electricity becomes greater whenthe price of its substitute good has become higher. The less available or the more expensive substitute goods are, the more it matters if the price of the good inquestionrises, and vice versa.
If, instead, gas and electricity happen to be complementary goods, a rise in the price of gas causes an inward shift of the demand curve for electricity. The addi-tional loss of consumer surplus of any concomitant rise in the price of electricity is thensmaller thanif the price of gas had not riseninthe first place. This also makes good sense, as an initial rise in the price of gas makes electricity less useful when it is complementary to gas – and not more useful as it will be when it is a substitute for gas.
4 The reader can soon convince himself that the analysis is symmetric for a sequential or simultaneous fall in the prices of two or more goods. A brief caveat is called for in this context because of the much-touted ‘path-dependence’problem which, when applied to the adding of consumer surplus, has it that the aggregate of consumer surpluses from several price changes will differ in general according to the order inwhich they are taken. Although the mathematical theorem is
Dg
Q2 Q1 g pg2
pg1
O Eg
Figure 5.3
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 35 — #4 Consumer surplus when prices change 35
e De
Q2 Q1
pe2
pe1
O Ee E9e
D9e
Figure 5.4
itself a valid one, it has no relevance to this particular economic exercise. The economist is obliged to take the number of price changes in a particular order only because he finds it convenient for calculation purposes to portray them within a partial equilibrium setting. These price changes are, however, deemed to occur simultaneously.
The imaginative reader may be able to picture a set of concave indifference surfaces in three-dimensional space, the vertical axis y being (real) income, the two horizontal axes being, respectively, goods x and z. A shift of the individual’s budget plane, arising from a simultaneous change in the prices of x and z, will touch only one of the indifference surfaces – a higher, lower, or the same one – at only one point. In consequence, there is a unique measure for any definition of consumer surplus.1
Were it possible, then, to imagine a set of n-dimensional indifference surfaces, a simultaneous change in any or all of the goods prices, represented now by a change in the n-dimensional plane, would again reveal a unique equilibrium and, therefore, a unique consumer surplus.
5 For expositional purposes, we have so far held supply prices of all goods constant. By now removing this simplification, we can see that the above analysis is applicable also to cases inwhich supply curves slope upward or downward. For,
1 Although, as affirmed inthe text, the theorem is without applicationto the simultaneous change ina number of goods prices, it is of passing interest to remark that the necessary and sufficient condition for path independence with respect to any pair of prices piand pjis that ∂qj/∂pi= ∂qi/∂pj(where qiand qjare the corresponding quantities), a condition that is explicit in the Hicksian system (see Hicks, 1939: Appendix).
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 36 — #5 if any good y is related to good x, the equilibrium price of y will also be affected if, in the first instance, there is an exogenous change in the price of x.
Let us restrict our attentionto the two-good case, inwhich the good that has an exogenous fall in price, say electricity, has constant costs and the related good, say gas, which is a substitute for electricity, does not have constant costs.
Again, using the device of taking the price changes in sequence, the exogenous fall inthe price of electricity from pe1 to pe2 first increases consumer surplus in electricity by the shaded area inFigure 5.5. But this fall inthe price of electricity induces a leftward shift of the demand curve for gas from DD to DDinFigure 5.6.
If we assume first that, as inFigure 5.6, gas has anupward-sloping supply curve, there will be a fall inthe equilibrium price of gas from pgto pg. In consequence, there will also be a small leftward shift inthe downward curve for electricity which, however, we provisionally ignore.
The total increment of welfare arising from the initial fall in the price of elec-tricity plus the further induced fall in the equilibrium price of gas is calculated by adding the shaded strip inFigure 5.6 to the shaded strip inFigure 5.5. The interpretation of this procedure is straightforward enough.
First, the shaded strip in Figure 5.5 represents the increment of consumer surplus arising from the fall in the price of electricity with the price of gas at pg. Second, the shaded strip in Figure 5.6 represents the further increment of consumer surplus for a fall inthe equilibrium price of gas from pgto pg with the price of electricity remaining at pe2. The sum of these two areas is thena measure of the amount that consumers are willing to pay for reducing the price of electricity from pe1 to pe2
when, as a result, the price of gas to them will also fall from pgto pg. Anextension of the analysis reveals that if gas has, instead, a downward-sloping supply curve, the leftward shift inits demand curve which is associated with the fall inthe price of electricity results in a higher equilibrium price for gas and therefore entails
De
e pe1
pe2
O Ee
Figure 5.5
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 37 — #6 Consumer surplus when prices change 37 D
O B
S
C g
D9
D
D9 pg
p9g
Figure 5.6
the loss of a strip of consumers surplus – consequently a subtraction of that strip from the shaded area in Figure 5.5. The resulting difference between the two areas is then a measure of the amount consumers are willing to pay when the price of electricity falls from pe1 to pe2and, as a result, the price of gas is increased.2
2 The measure of simultaneous changes, whether of prices or availabilities, can have particular importance when measuring the community’s loss or benefit from alterations in the amounts of collective goods or bads. To illustrate with a case of related sources of disamenity, say two chief sources of noise ina givenarea, that from cars and that from aircraft.
For each source of noise, we have an individual’s downward-sloping marginal valuation curve which measures the maximum sum he would pay to be rid of successive units of noise, beginning with some almost unbearable volume of noise, given – and this is critical – the existing large volume of noise from the other source.
With this sort of ceteris paribus, consider the marginal valuation curve for reducing car noise.
Since there is not that much benefit to the individual from reducing car noise while aircraft noise continues at its high level, such a marginal valuation curve would not be very high. The same is true of the marginal valuation curve for reductions in aircraft noise.
Obviously, the benefit to the individual of a simultaneous reduction in the noise of both would be considerable: certainly more than the sum of the valuation as measured by the areas under the two ceteris paribus marginal valuation curves.
The correct measure of the benefit of removing both sources of noise is derived by adding to the benefit (or consumer surplus) as measured under the marginal valuation curve for car-noise riddance given the existing high volume of aircraft noise, the subsequent benefit (or consumer surplus) as measured by the area under the now much higher marginal valuation curve for aircraft-noise reduc-tion. This latter curve is now much higher, simply because the relevant ceteris paribus contains the information that all car noise has been removed. Consequently, removal of aircraft noise now does make a real difference to the individual’s amenity.
QUAH: “CHAP05” — 2007/1/25 — 19:06 — PAGE 38 — #7 Symmetrical reasoning applies also to goods that are complements. If, for example, gas were now complementary with electricity, a fall in the price of elec-tricity would cause an outward shift inthe demand curve for gas. First, assume the supply curve of gas to be upward sloping. The outward shift in its demand curve increases the equilibrium output and price of gas. This rise in the price of gas entails a consumer loss which must be subtracted from the consumer gain arising from the initial fall in the price of electricity.
If, conversely, the supply curve of gas is downward sloping, the outward shift in its demand curve, arising from the initial fall in the price of electricity, results in a fall in the output and equilibrium price of gas. To that extent, there is now an additional consumer surplus in gas to be added to the increase in consumer surplus from the fall inthe price of electricity.
6 To introduce a little more complication, let the supply curves slope upward both for electricity and gas. It now follows that the induced fall in the price of gas (from the inward shift of its demand curve resulting from the reduced price of electricity) will itself induce an inward shift in the demand curve for electricity, and therefore a fall inits price. Some further correctioninconsumer surplus is therefore necessary.
One may continue in this way indefinitely, although under plausible assumptions (related to familiar stability conditions) these mutually induced shifts in the two demand curves become smaller and converge to new equilibrium prices for both gas and electricity.
However, since the errors in estimating the relevant demand curves – in the above example, the initial demand curve for electricity and the inwardly shifted demand curve for gas, following a fall in the price of electricity – are likely to be large enough to swamp the refinements from further mutually induced shifts in the demand curves, some attempted corrections are best ignored. They would be worthwhile only if the initial price change was unusually large.
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