Question

(20 points) Suppose the government wishes to tax a utility maximizing consumer to obtain a certain amount of tax revenue. A utility maximizing consumer has utility function ?(?, ?) = √? + ?. The price of ? is $1, the price of ? is $4 and the consumer’s income is $120. (a) Suppose the government imposes sales tax ? = 1 on good ? per unit. What is the optimal consumption for good ? and good ? for the consumer under the sales tax? What is the utility level that the consumer achieves under the sales tax? How much revenue does the government collect by imposing the sales tax? (b) Suppose the government imposes income tax instead of sales tax to collect the same amount of revenue that could be earned from imposing sales tax in part (a). What is the optimal consumption for good ? and good ? for the consumer under the income tax? What is the utility level that the consumer achieves under the income tax? Compare the utility level under the income tax with the utility level under the sales tax in part (a).

Answer 1

utility function a ulo, y) = $1 ty P2= $1; Py = 44 ;. Income (m) = $120. a) Budget constraints Alla typy = m ..... "max : V = 3a +y ...@ At optimal. Slobe of Tridifference - slope el, Budget curve ia on these porno 1 64.5=x)... substituting ② in budget constraint a Pi + y Py = M N y la + y Py = m 2 upy - Mche mi B 's 1+ i tzi un god tim So; effective prices on 2] B; optime * : [v-) (line) 1.3 [prima di tutte le 20 - 4324-5 ] abhin das [= 1 ; g = 29.5]

2121 X nye im Nov Do Jaty" - JT +29:5' = 30.5 Tax Revenue = trus 1:1 =$1 by now Income tax equal to tax Revenue in parit kn) it Applied . new Income (m²) = M-1 = 120-1 = 119 : - Px = $1; Py = 84 **(49) (03) 4 99 - E - 24.757 [9 * = 28.757 U- Jaa+y9= 14 + 28.75= 2+28.75 [= 30.75 ) So, utility under in come tax is greater than . the sales tax for same tax revenue. i = 28.75