# Why does lump sum tax decrease budget line

I only need help with (f). A hint would be appreciated.

Problem: Donald derives utility from only two goods, carrots (Qc) and donuts (Qd). His utility function is as follows: U(Qc,Qd) = (Qc)(Qd)

The marginal utility that Donald receives from carrots (MUc) and donuts (MUd) are given as follows: MUc = Qd MUd = Qc

Donald has an income (I) of 120 dollars and the price of carrots (Pc) and donuts (Pd) are both $1. a. What is Donald's budget line? b. What is Donald's income-consumption curve? c. What quantities of Qc and Qd will maximize Donald's utility? d. Holding Donald's income and Pd constant at 120 dollars and$1 respectively, what is Donald's demand curve for carrots?

e. Suppose that a tax of $1 per unit is levied on donuts. How will this alter Donald's utility maximizing market basket of goods? f. Suppose that, instead of the per unit tax in (e), a lump sum tax of the same dollar amount is levied on Donald. What is Donald's utility maximizing market basket? • Do you know what "lump sum tax" means? Sep 28, 2022 at 7:57 • I think you might have missed some information provided in the problem. does not make much sense to levy a 1usd tax per unit while each unit costs 1usd. Also, a lump sum tax on consumer expenditure would decrease the budget by the lump sum tax amount. so that should be$120 - lumpsumtax = Qc + Qd\$. But why do you have 90, are you sure the lump sum tax to the consumer expenditure is not 30? Sep 28, 2022 at 7:59
• @Giskard How ever many donuts you choose to buy is the tax you will have to pay. Could you please give a hint on how to solve part f? Sep 28, 2022 at 8:19
• @Macosso I added in the complete problem. Sep 28, 2022 at 8:19

## 1 Answer

Hint for point (f):

(f). Suppose that, instead of the per unit tax in (e), a lump sum tax of the same dollar amount is levied on Donald. What is Donald's utility maximizing market basket?

I think that the point (f) asks you to calculate preliminarily the total amount of taxes paid in point (e). To do this you have to multiply the unit tax of of $$1$$ dollar for the quantity of donuts in the basket that maximize utility in this case, let's call this quantity $$Q_d^*$$. This amount of $$Q_d^*$$ is the lump sum tax you have to consider in point (f).

To solve point (f), this $$Q_d^*$$ must be then subtracted from the income of the consumer, and with this new budget constraint (the budget constraint shifts downward) you can calculate the new quantity of donuts (and of carrots) in the basket that now maximizes utility.

Basically, point (f) is a question about how different kinds of taxation, indirect taxation or lump sum taxes, the amounts being equal, can affect the quantity demanded of a good.