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Chapter Fourteen Consumer’s Surplus

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Monetary Measures of Gains-to- Trade Suppose you know you can buy as much gasoline as you choose at a given price of $1 per gallon once you have entered the gasoline market. Q: What is the most you would pay to be able to enter the market for gasoline?

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A: You would pay a sum up to, but not exceeding, the dollar value to you of the gains-to-trade you would enjoy once inside the market. How can we measure the monetary values of such gains-to-trade? Monetary Measures of Gains-to- Trade

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We will consider three such measures Consumer’s Surplus Equivalent Variation, and Compensating Variation. Only in one special circumstance do these three measures coincide. Monetary Measures of Gains-to- Trade

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Suppose gasoline is purchasable only in lumps of one gallon and consider a single consumer. Ask “What is the most she would pay for a 1st gallon?”. Call this r 1, her reservation price for the 1st gallon. r 1 is the dollar equivalent of the marginal utility of the 1st gallon. $ Equivalent Utility Gains

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Now that she has one gallon, ask “What is the most she would pay for a 2nd gallon?”. Call this r 2, her reservation price for the 2nd gallon. r 2 is the dollar equivalent of the marginal utility of the 2nd gallon. $ Equivalent Utility Gains

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More generally, if she already has n-1 gallons of gasoline then let r n be the most she will pay for an nth gallon. r n is the dollar equivalent of the marginal utility of the nth gallon. $ Equivalent Utility Gains

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The sum r 1 + … + r n will therefore be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of $0. So r 1 + … + r n - p G n will be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of $p G each. $ Equivalent Utility Gains

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A plot of r 1, r 2, …, r n, … against n is a reservation-price curve. This is not quite the same as the consumer’s demand curve for gasoline. $ Equivalent Utility Gains

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123456 r1r1 r2r2 r3r3 r4r4 r5r5 r6r6

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What is the monetary value of our consumer’s gain-to-trading in the gasoline market at a price of $p G ? $ Equivalent Utility Gains

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For the 1st gallon the dollar equivalent of the net utility gain is $(r 1 - p G ). For the 2nd gallon the dollar equivalent of the net utility gain is $(r 2 - p G ). And so on, so the monetary value of the gain-to-trade is $(r 1 - p G ) + $(r 2 - p G ) + … for as long as r n - p G > 0. $ Equivalent Utility Gains

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123456 r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 pGpG

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123456 r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 pGpG $ value of net utility gains-to-trade

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Now suppose that gasoline is sold in half-gallon units. Now let r 1, r 2, …, r n, … denote the consumer’s reservation prices for successive half-gallons of gasoline. Our consumer’s new reservation price curve is $ Equivalent Utility Gains

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123456 r1r1 r3r3 r5r5 r7r7 r9r9 r 11 7891011

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$ Equivalent Utility Gains 123456 r1r1 r3r3 r5r5 r7r7 r9r9 r 11 7891011 pGpG

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$ Equivalent Utility Gains 123456 r1r1 r3r3 r5r5 r7r7 r9r9 r 11 7891011 pGpG $ value of net utility gains-to-trade

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Suppose gasoline can be purchased in any quantity. Then our consumer’s reservation price curve is $ Equivalent Utility Gains

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Gasoline ($) Res. Prices pGpG Reservation Price Curve for Gasoline $ value of net utility gains-to-trade

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If we approximate the net utility gain area under the reservation-price curve by the corresponding area under the ordinary demand curve then we get the Consumer’s Surplus approximate measure of net utility gain from buying gasoline at a price of $p G. Consumer’s Surplus

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Consumer’s Surplus is an exact dollar measure of total utility gains from consumption of commodity 1 when the consumer’s utility function is quasilinear in commodity 2. Otherwise Consumer’s Surplus is an approximation. Consumer’s Surplus

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The change to a consumer’s total utility due to a change to p 1 is approximately measured by the change in her Consumer’s Surplus. Consumer’s Surplus

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p1p1 For quasi-linear preferences, p 1 (x 1 ) is the inverse ordinary demand curve for commodity 1

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Consumer’s Surplus p1p1 CS before p 1 (x 1 )

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Consumer’s Surplus p1p1 CS after p 1 (x 1 )

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Consumer’s Surplus p1p1 Lost CS p 1 (x 1 ), inverse ordinary demand curve for commodity 1.

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Two additional monetary measures of the total utility change caused by a price change are Compensating Variation and Equivalent Variation. Compensating Variation and Equivalent Variation

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Suppose the price of commodity 1 rises. Q: What is the smallest amount of additional income which, at the new prices, would just restore the consumer’s original utility level? A: The Compensating Variation. Compensating Variation

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x2x2 x1x1 u1u1 p1=p1’p1=p1’p 2 is fixed.

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Compensating Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed.

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Compensating Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed.

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Compensating Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed. CV = m 2 - m 1.

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Suppose the price of commodity 1 rises. Q: What is the smallest amount of additional income which, at the original prices, would just restore the consumer’s original utility level? A: The Equivalent Variation. Equivalent Variation

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x2x2 x1x1 u1u1 p1=p1’p1=p1’p 2 is fixed.

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Equivalent Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed.

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Equivalent Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed.

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Equivalent Variation x2x2 x1x1 u1u1 u2u2 p1=p1’p1=p1”p1=p1’p1=p1” p 2 is fixed. EV = m 1 - m 2.

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What relationships exist between the three monetary measures of utility changes due to price changes? Relationship 1: When the consumer’s preferences are quasilinear, all three measures are the same. Consumer’s Surplus, Compensating Variation and Equivalent Variation

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Changes in the welfare of a firm can be measured in monetary units in much the same way as for a consumer. Producer’s Surplus

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y (output units) Output price (p) Marginal Cost

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Producer’s Surplus y (output units) Output price (p) Marginal Cost Revenue =

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Producer’s Surplus y (output units) Output price (p) Marginal Cost Variable Cost of producing y’ units is the sum of the marginal costs

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Producer’s Surplus y (output units) Output price (p) Marginal Cost Variable Cost of producing y’ units is the sum of the marginal costs Revenue less VC is the Producer’s Surplus.

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