Question

Using the simplified one-period model explain in words and illustrate via a diagram how moving from a proportional tax to a lump sum tax would increase consumer welfare. Explain why the lump sum tax is more efficient.

Answer 1

a proportional tax changes the price of the good.

say two goods x and y, U(x,y) is the utility function that gives convex ICs. p1 and p2 be the price of x and y respectively. m is income

say a proportional tax of t on good x is intoduced. it will change the price of x from p1 to p1+t. thus the budget line pivots about the x-axis(good x). A price changes have a dual effect on a consumer’s otpimal demand functions which are substitution effect, and the income effect.earlier, at utility maximising bundle, MRS=p1/p2. now, MRS = (p1+t)/p2, thus the U-maximising budle changes from a on U0 to B on U1.

but with a lumpsum tax of s on income, income changes while the price ratio is same. When income changes, there is only one effect – the income effect. and the equilibrium condition remains as MRS=p1/p2. individual shifts from A on U0 to C on U3.

A deadweight loss arises in case of proportional tax beacuse consumers substitute their consumption away from x to y, which is now relatively cheaper than x. but the lump sum tax does not induce this kind of behavior (i.e. there is no substitution effect with a lump sum tax). both x and y decrease.

see the graph. U3 is higher than U2. so the welfare or utility reduced is more under proportional tax (U2) than under lumpsum tax (U3), that is why lumpsum taxes are more efficient.