Chapter Fourteen Consumer’s Surplus

Monetary Measures of Gains-to- Trade  You can buy as much gasoline as you wish at $1 per litre once you enter the gasoline market.  Q: What is the most you would pay to enter the market?

 A: You would pay up to the dollar value of the gains-to-trade you would enjoy once in the market.  How can such gains-to-trade be measured? Monetary Measures of Gains-to- Trade

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

 Suppose gasoline can be bought only in lumps of one litre  Use r 1 to denote the most a single consumer would pay for a 1st litre -- call this his reservation price for the 1st litre  r 1 is the dollar equivalent of the marginal utility of the 1st litre Dollar Equivalent of a Utility Gain

 Now that he has one litre, use r 2 to denote the most he would pay for a 2nd litre -- this is his reservation price for the 2nd litre  r 2 is the dollar equivalent of the marginal utility of the 2nd litre $ Equivalent of Utility Gains

 Generally, if he already has n-1 litres of gasoline then r n denotes the most he will pay for an nth litre.  r n is the dollar equivalent of the marginal utility of the nth litre. Dollar Equivalent of Utility Gains

 r 1 + … + r n is therefore the dollar equivalent of the total change to utility from buying n litres of gasoline at a price of $0.  So r 1 + … + r n - p G n is the dollar equivalent of the total change to utility from acquiring n litres of gasoline at a price of $p G each. $ Equivalent of Utility Gains

 The graph of r 1, r 2, …, r n, … against n is a reservation-price curve  Note : This is generally not quite the same as the consumer’s demand curve for gasoline (see later) $ Equivalent of Utility Gains

$ Equivalent Utility Gains r1r1 r2r2 r3r3 r4r4 r5r5 r6r6

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

 The dollar equivalent net utility gain for the 1st litre is $(r 1 - p G )  and is $(r 2 - p G ) for the 2nd litre,  and so on, so the dollar 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

r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 pGpG

r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 pGpG

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

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

$ Equivalent of Utility Gains r1r1 r3r3 r5r5 r7r7 r9r9 r

$ Equivalent of Utility Gains r1r1 r3r3 r5r5 r7r7 r9r9 r pGpG

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

 And if gasoline is available in one- quarter litre units... $ Equivalent Utility Gains

$ Equivalent Utility Gains pGpG

$ Equivalent Utility Gains pGpG $ value of net utility gains-to-trade

 Finally, if gasoline can be purchased in any quantity then... $ Equivalent Utility Gains

Gasoline ($) Res. Prices Reservation Price Curve for Gasoline

$ Equivalent Utility Gains Gasoline ($) Res. Prices pGpG Reservation Price Curve for Gasoline

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

 Estimating a consumer’s reservation- price curve is difficult  So, as an approximation, the reservation-price curve is replaced by the consumer’s ordinary demand curve $ Equivalent Utility Gains

 A consumer’s reservation-price curve is not quite the same as his ordinary demand curve. Why not?  A reservation-price curve describes sequentially the values of successive single units of a commodity  An ordinary demand curve describes the most that would be paid per unit for q units of a commodity, purchased simultaneously Consumer’s Surplus

 Approximating the net utility gain area under the reservation-price curve by the corresponding area under the ordinary demand curve gives the Consumer’s Surplus measure of net utility gain Consumer’s Surplus

Gasoline ($) Reservation price curve for gasoline Ordinary demand curve for gasoline

Consumer’s Surplus Gasoline Reservation price curve for gasoline Ordinary demand curve for gasoline pGpG ($)

Consumer’s Surplus Gasoline Reservation price curve for gasoline Ordinary demand curve for gasoline pGpG $ value of net utility gains-to-trade ($)

Consumer’s Surplus Gasoline Reservation price curve for gasoline Ordinary demand curve for gasoline pGpG $ value of net utility gains-to-trade Consumer’s Surplus ($)

 The difference between the consumer’s reservation-price and ordinary demand curves is due to income effects  But, if the consumer’s utility function is quasilinear in income, then there are no income effects and Consumer’s Surplus is an exact $ measure of gains-to-trade Consumer’s Surplus

 Remember: When the consumer’s utility function is quasilinear in income, the indifference curves (with income on the vertical axis) are vertical displacements of the same curve; as income increases (holding prices constant) there is no change in the demand for x 1 Consumer’s Surplus

Consumer’s Surplus with QL U fn The consumer’s utility function is quasilinear in x 2. Take p 2 = 1. Then the consumer’s choice problem is to maximize subject to

Consumer’s Surplus with QL U fn The consumer’s utility function is quasilinear in x 2. Take p 2 = 1. Then the consumer’s choice problem is to maximize subject to

Consumer’s Surplus with QL U fn That is, choose x 1 to maximize The first-order condition is That is, This is the equation of the consumer’s ordinary demand for commodity 1

Consumer’s Surplus with QL U fn Ordinary demand curve, p1p1 CS

Consumer’s Surplus with QL U fn Ordinary demand curve, p1p1 CS

Consumer’s Surplus with QL U fn Ordinary demand curve, p1p1 CS

Consumer’s Surplus with QL U fn Ordinary demand curve, p1p1 CS is exactly the monetary value of the consumer’s net utility gain from consuming x 1 ’ units of good 1

 Consumer’s Surplus is an exact dollar measure of utility gained from consuming commodity 1 when the consumer’s utility function is quasilinear in commodity 2  Otherwise Consumer’s Surplus is an approximation Summary: Consumer’s Surplus

It follows that  The change in a consumer’s total utility due to a change in p 1 is approximately the change in his Consumer’s Surplus Consumer’s Surplus when price changes

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

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

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

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

 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

 p 1 rises  Q: What is the minimum extra income that, at the new prices, just restores the consumer’s original utility level?  That is, how much extra income is needed to ‘compensate’ the consumer for the price change? Compensating Variation

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

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

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

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

 p 1 rises  Q: What is the maximum income reduction that, at the original prices, would allow the consumer to just reach the utility level he enjoys after the price change?  That is, how much income would you have to take away from him, before the price change, to reduce his utility to the level reached after the price change Equivalent Variation

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

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

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

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

 Note that: The CV of a price increase is the EV of the (reverse) price decrease, and the EV of a price increase is the CV of the (reverse) price decrease For a price increase, CV > EV. For a price decrease, EV > CV Consumer’s Surplus, Compensating Variation and Equivalent Variation

 Relationship 1: When the consumer’s preferences are quasilinear, all three measures are the same. Consumer’s Surplus, Compensating Variation and Equivalent Variation

So when the consumer has quasilinear utility, CV = EV =  CS. But, otherwise, we have: Relationship 2: In size, For a price increase, EV <  CS < CV For a price decrease, CV <  CS < EV

 Changes in a firm’s welfare can be measured in dollars much as for a consumer Producer’s Surplus

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

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

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

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

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

Interpetation of Producer’s Surplus  Producers’surplus = total revenue earned by producing x* minus variable cost of producing x*. Therefore, the PS includes a. fixed costs b. ‘pure profit’ (if any)  In agriculture, PS covers the costs of fixed factors (land, family labour)

Interpetation of Producer’s Surplus  In agriculture, PS covers the costs of fixed factors (land, family labour) plus (possibly) positive profit  But if positive profit is earned, this means that farms want to grow, and the price of land rises until, in the long run, all PS is absorbed by the fixed factors

END OF LECTURE  End of lecture  The following slides are for self-study

Consumer’s Surplus, Compensating Variation and Equivalent Variation Ifthen

Consumer’s Surplus, Compensating Variation and Equivalent Variation Ifthen and so the change in CS when p 1 rises from p 1 ’ to p 1 ” is

Consumer’s Surplus, Compensating Variation and Equivalent Variation Ifthen and so the change in CS when p 1 rises from p 1 ’ to p 1 ” is

Consumer’s Surplus, Compensating Variation and Equivalent Variation Ifthen and so the change in CS when p 1 rises from p 1 ’ to p 1 ” is

 Now consider the change in CV when p 1 rises from p 1 ’ to p 1 ”.  The consumer’s utility for given p 1 is and CV is the extra income which, at the new prices, makes the consumer’s utility the same as at the old prices. That is,... Consumer’s Surplus, Compensating Variation and Equivalent Variation

So

 Now consider the change in EV when p 1 rises from p 1 ’ to p 1 ”.  The consumer’s utility for given p 1 is and EV is the extra income which, at the old prices, makes the consumer’s utility the same as at the new prices. That is,... Consumer’s Surplus, Compensating Variation and Equivalent Variation

That is,