Print this page out Use the space below and the back to writ

Print this page out. Use the space below and the back to write out and clearly indicate your answers. Do not use this sheet as scrap paper, but use it to neatly present your work. Consider Logan, a consumer who has preferences represented by the utility function U(H,/) = H2 +/3, where H represents the number of healthy meals Logan consumes per week and J represents the number of unhealthy junk) meals that Logan consumes per week. Logan has an income of $42 per week. The price of healthy meals and unhealthy meals are each $2 per meal. Find Logan\'s utility-maximizing consumption bundle. What proportion of the meals that Logan consumes are healthy? Now suppose that Logan\'s parents offer to pay for 50% of Logan\'s healthy meals, thereby lowering the price of a healthy meal to $1. Find Logan\'s new utility-maximizing consumption bundle. Now what proportion of the meals that Logan consumes are healthy?

Solution

Consider the given problem here the typical consumer “Logan” have a utility function, “U=H^2/3 + J^2/3”.

So, this utility function is a convex to the origin with negatively sloped, => the optimum consumption bundle will be determined by the condition, “MRS = Ph/Pj=1”, where “Ph=Pl=$2”.

So, given the utility function, “U=H^2/3 + J^2/3”, the “Marginal Utility of H” is MUh = 2/3*H^(-1/3), and the “marginal utility of J” is MUj = 2/3*J^(-1/3).

So, the MRS=MUh/MUj = 2/3*H^(-1/3)/2/3*J^(-1/3) = H^(-1/3)/J^(-1/3) = (J^1/3)/(H^1/3) = (J/H)^1/3.

So, at the optimum, MRS = Ph/Pj =1, => (J/H)^1/3 = 1, => J = H.

Now, the budget line is given by, “Ph*H + Pj*J = M = 42, => 2*H+2*J = 4*H = 42, => H=42/4=10.5.

So, at the given price and income the optimum choice of the “H” and “J” is “H=J=10.5”.

So, the proportion of health meals is “H/J=10.5/10.5=1”.

b).

Now, suppose that “Logan’s” parents offer to pay for 50% of Logan’s healthy meals, => “Ph” will lower to “$1” form $2. So, the now the equation of new budget constraint is given by, “H + 2*J = 42”.

So, now at the optimum MRS = Ph/Pj = ½ , => (J/H)^1/3 = ½ , => J/H=1/8, => H = 8*J.

Now, substituting this condition into the new budget line we will get the optimum solution for “H” and “J”.

=> H + 2*J = 42, => 8*J + 2*J = 10*J = 42, => J=42/10 = 4.2, => H=8*J=8*4.2=33.6.

So, here the optimum choice is “H=33.6” and “J=4.2”, the proportion of healthy meals is “H/J = 33.6/4.2 = 8” (got from “MRS=Ph/Pj condition).