A consumer with the utility function U(x,, x,) = xZx3 faces prices p, = 4, p, = 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;. 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. Consider the consumer in (2), but ignore the information in (3). Holding prices fixed: a. Derive an expression for the Engel Curve for x,. Hint: use the demand function to get a relationship between m and x; (the optimal demand for x5), holding p, and p, fixed. Plot this Engel curve (put m on the vertical axis, and x; on the horizontal axis). c. On a different graph, plot this consumer's demand curve for x,, holding p, and m fixed. Label the axes