3. (6 pts total) Diego's utility function is $U(q_1, q_2) = 2 \left(q_1 \frac{3}{2} q_2 \frac{1}{2} \right)$, where $q_1$ is chocolate bars and $q_2$ is slices of pie. If $P_1 = $1, and $P_2 = $2, and his budget is $Y = $40, what is his optimal bundle of the two goods?
a. (1 pt) First, set up the Lagrangian for is optimization problem:
b. (1 pt)) Second, derive the first order conditions (there should be three equations).
c. (1 pt) Finally, solve for the optimal $q_1$ and $q_2$. (Box your final answer, but additionally show your work).
For fun, lets also walk through how to solve this problem with the book's \"short cut\":
d. (1 pt) First, find $MRS_{1,2}$ given the previously stated model assumptions.
e. (1 pt) Second, find $MRT_{1,2}$ given the previously stated model assumptions.
f. (1 pt) Finally, use answers to d and e, in addition to the budget constraint, and find the optimal $q_1$ and $q_2$. (Box your final answer, but additionally show your work. State where your work can be found if you need more space).
