Consumption -Applications Rates of Change Mathematically, the Slutsky decomposition can be written in the following way. Ax1=△r1+△r1+△r · What is the rate of change of demand with respect to price,△r1/△ Suppose price changes from Pf to pi. The old and new amounts of money income required to a"=Pr1+p2I2 and m= PI1+paI2 respectively. Subtracting one from the other gives th quired to enable the just afford bundle z =(r1I2) · For convenience, define△r1=-△r1, the negative of the income effect.△r1/△ m is then the change in demand hen income changes.So△r"/Ap=-△m/△p1=-x1△r/△m. The endowment income effect is given by the change in demand when income changes multiplied by the change in acome when price changes. The first term is Ar" /Am · The second term is△m/△P1.Now,m4=P1+p22andm=1+p2u2.So△m=m2-m=14p Therefore△m/△p1=a1. Finally Ar1△xi,△x,A式△x ApIApI 4p,* ①P +(a1-x1) △m This is the standard way to write the slutsky equation usually with differentials. It can be applied easily. Consumption-Applicatiorsa Tax rebates The government decides to tax beer. They return all the revenue to the consumer making them no worse off The amount of beer bought before the tax was b. The amount of other consumption Make 'other consumption'the numeraire. The price of beer is p. Adding a tax to the price of beer effectively increases the price the consumer faces to p+t. The consumer will change consumption due to the price change and due to their increased income from the tax rebate. Suppose the new bundle is(0, c). The government raises th in tax. Income before the tax was m. After the tax rebate they will get m +tb. The budget line before the tax (p+t)b +e. This equation simplifies to m= pb+c So bundle(b, e)was affordable when the consumer chose bundle(6, c). They cannot prefer it and must be worse off.Consumption — Applications 3 Rates of Change • Mathematically, the Slutsky decomposition can be written in the following way. ∆x1 = ∆x s 1 + ∆x n 1 + ∆x ω 1 • What is the rate of change of demand with respect to price, ∆x1/∆p1? • Suppose price changes from p a 1 to p b 1 . The old and new amounts of money income required to purchase x are ma = p a 1x1 + p2x2 and mb = p b 1x1 + p2x2 respectively. Subtracting one from the other gives the change in money income required to enable the consumer to just afford bundle x = (x1, x2): ∆m = mb − ma = x1(p b 1 − p a 1 ) = x1∆p1 • For convenience, define ∆x m 1 = −∆x n 1 , the negative of the income effect. ∆x m 1 /∆m is then the change in demand when income changes. So ∆x n 1 /∆p1 = −∆x m 1 /∆p1 = −x1∆x m 1 /∆m. • The endowment income effect is given by the change in demand when income changes multiplied by the change in income when price changes. The first term is ∆x m 1 /∆m. • The second term is ∆m/∆p1. Now, ma = p a 1ω1 + p2ω2 and mb = p b 1ω1 + p2ω2. So ∆m = mb − ma = ω1∆p1. Therefore ∆m/∆p1 = ω1. Finally: ∆x1 ∆p1 = ∆x s 1 ∆p1 + ∆x n 1 ∆p1 + ∆x ω 1 ∆p1 = ∆x s 1 ∆p1 + (ω1 − x1) ∆x m 1 ∆m • This is the standard way to write the Slutsky equation — usually with differentials. It can be applied easily. Consumption — Applications 4 Tax Rebates • The government decides to tax beer. They return all the revenue to the consumer — making them no worse off? • The amount of beer bought before the tax was b. The amount of other consumption was c. Make ‘other consumption’ the numeraire. The price of beer is p. Adding a tax to the price of beer effectively increases the price the consumer faces to p + t. • The consumer will change consumption due to the price change and due to their increased income from the tax rebate. Suppose the new bundle is (b 0 , c 0 ). The government raises tb0 in tax. • Income before the tax was m. After the tax rebate they will get m + tb0 . The budget line before the tax was m = pb + c. After the tax it is m + tb0 = (p + t)b 0 + c 0 . This equation simplifies to m = pb0 + c 0 . • So bundle (b 0 , c 0 ) was affordable when the consumer chose bundle (b, c). They cannot prefer it and must be worse off