A consumer with the utility function $U(x_1,x_2)=x_1^2{x}_2^3$ faces prices $p_1=4,p_2=5$ and

has an income of $$200.$ Compute the effect of an infinitesimally small increase in

income on the consumer's maximized utility. Hint: Lagrange multiplier.

Consider the consumer in $(2).$ Suppose a tax of $$1$ per unit is imposed on the

consumption of $x_1.$

a$.$ How much tax revenue would be raised as result?

b$.$ What is the consumer's utility level after the tax?

c$.$ If the same tax revenue was raised via a lumpsum income tax, leaving prices

unchanged, compute the consumer's utility level after the lumpsum tax.