Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y represent the quantities of two goods, X and Y.

Now assume that the consumer has $31 to spend on goods X and Y, which have fixed prices of pX=3 and pY=4.

(d) Express the consumer’s budget constraint as an equation. The only variables in the equation shouldbe x and y.

(e) Calculate the first-order condition for the consumer’s optimization problem of dividing her money between goods X and Y in the way that maximizes her utility. (The only variables in the equationshould be x and y.)

(f) Calculate the consumption bundle (x,y) that solves the first-order condition for the consumer’s problem.

(g) Calculate the two consumption bundles (x,y) that are on the boundary of the consumer’s optimization problem.

(h) For the three possible solutions that you found in parts (f) and (g), calculate the utility that the consumer would receive from each bundle. Whatshould she do to maximize her utility? Justify your answer carefully.

(i) Repeat parts (f), (g), and (h), assuming that, ceteris paribus, the price of X rises to pX=$75.

(j) What should the consumer do to maximize her utility if, ceteris paribus, the price of X rises topX=100? Justify your answer carefully.

(k) Compare your answers to parts (h), (i), and (j). Discuss the connection between these answers and the Law of Demand.