# Chapter 5 Some Applications of Consumer Demand, and Welfare Analysis.

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Chapter 5 Some Applications of Consumer Demand, and Welfare Analysis

Price Sensitivity of Demand  Elasticity of demand Percentage change in demand From a given percentage change in price 2

Price-Elasticity Demand Curves  Elastic demand, |ξ|>1 1% change in price ○ >1% change in quantity demanded  Inelastic demand, |ξ|<1 1% change in price ○ <1% change in quantity demanded  Unit elastic demand, |ξ|=1 1% change in price ○ =1% change in quantity demanded 3

Elasticity along a linear demand curve 4 Quantity 0 Price A P max q= a-b.p μ |ξ|<1 |ξ|>1 P1P1 |ξ|=1

Price-Elasticity Demand Curves  Perfectly inelastic demand curve Perfectly vertical demand curve Zero quantity response to a price change  Perfectly elastic demand curve Horizontal demand curve Price > p ○ Quantity = 0 Price = p ○ Any quantity 5

Perfectly elastic & perfectly inelastic demand curves 6 Perfectly inelastic demand curve. With zero elasticity, the quantity demanded is constant as prices change. Quantity 0 Price D (a) Perfectly elastic demand curve. With infinite elasticity, the quantity demanded would be infinite for any price below p and zero for any price above p. Quantity 0 Price D (b)

Properties of Demand Functions 1. Price and income multiplication by the same factor leaves demand unaffected “No money illusion property” 7

1. No Money Illusion Property 8 Multiplying all prices by the same factor shifts the budget line from BB’ to B’”B”. Multiplying prices and the agent’s income by the same factor has no effect on the budget line. Good 1 ( x 1 ) 0 Good 2 ( x 2 ) e f B B’ B” B’”

2. Ordinal utility property 9 Regardless of the utility numbers assigned to the three indifference curves, the agent maximizes utility by choosing point e. Thus demand is unaffected Good 1 ( x 1 ) 0 Good 2 ( x 2 ) B’ B 90(3) 100(5) 120(8) e

From Individual to Market Demand  Market demand curve Aggregate of individual demand curves Horizontally add up individual demand curves 10

Market demand from individual demand 11 (a) Person i Quantity 5 13 P1P1 P2P2 Price (b) Person j Quantity 10 20 Price (c) Person k Quantity 12 30 Price (d) Aggregate demand Quantity 27 63 Price DiDi DjDj DkDk The market demand curve D is the horizontal summation of the individual demand curves D i, D j, and D k. D

Welfare Measures  The welfare effects of price increase can be assessed using Demand curve: ○ Loss in consumer surplus Consumer choice model: ○ Price compensating variation

1. Consumer Surplus Consumer surplus Net gain to consumers measured as the difference between the willingness to pay and the amount actually paid 13

14 Quantity of cocaine demanded 0 Price 70 10 33.3 100 CS 1. Consumer Surplus Consumer surplus. The area under the demand curve and above the price measures the agent’s total willingness to pay for the quantity of the good she is consuming minus the amount she must pay.

Measures of Consumer Gain/ Loss  Loss of consumer surplus Difference between ○ consumer surplus for price p ○ consumer surplus for price p+∆p 15

Change in consumer surplus 16 When the price increases, the change in the area under the demand curve and above the price measures the welfare loss caused by the price change. Good 1 ( x 1 ) 0 Price p a p+∆p

2. Price-compensating variation in income  Price-compensating variation in income measures the compensation needed due to an increase in price  To understand the price-compensating variation in income we first introduce the expenditure function  Expenditure function Minimum income/expenditure amount (E) To achieve a predetermined utility (u) At given prices (p 1,p 2 )  E=E(p 1,p 2,u) 17

The Expenditure Function  The problem  The Lagrangian 18

Derivation of an Expenditure function 19 Suppose p1=\$0.5 and P2=\$1, What is the minimum level of income needed to bring the consumer to a utility level of u*? Good 1 ( x 1 ) 0 Good 2 ( x 2 ) I 1 (u*) f e B1B1 10 15 7 20 B2B2 B3B3 17

Measures of Consumer Gain/ Loss  Price-compensating variation in income Additional income given to consumer After price change Same utility (before price change) 20

Price-compensating variation in income 21 ZB is the amount of income that must be given to the agent after the price increase in order to restore him to I 1, the indifference curve he was on before the price change Good 1 ( x 1 ) 0 Good 2 I1I1 I2I2 f e d B” Z B’ B p Price -compensating variation (in income) Suppose p1=\$1 and P2=\$1. If P2 increases to \$2, How much extra income is needed to compensate the consumer?

Price-Compensating Variations and Expenditure Functions  Prices: p 1, p 2 Utility level: u* Expenditure: E=E(p 1,p 2,u*)  Increase in p 1 to p 1 + Expenditure: E’=E(p1+,p2,u*)  Price-compensating variation = E’-E= = E(p1+,p2,u*) - E(p 1,p 2,u*) 22

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