## Q1
Suppose Marvin has utility function $u\left(q_{1}, q_{2}\right)=q_{1}^{1 / 2} q_{2}^{1 / 2}$.

### a
Find Marvin's demand for $q_{1}$ and $q_{2}$ in terms of $Y, p_{1}$, and $p_{2}$.

### b
Marvin has income $Y=100$ and initially faces prices $p_{1}=1$ and $p_{2}=2$. What are his optimal quantities of $q_{1}$ and $q_{2}$ ?

### c

The price of $q_{1}$ increases to 2 . What are his new optimal quantities? What is the compensating variation?

## Q2
Suppose Lucy has quasilinear utility $u\left(q_{1}, q_{2}\right)=2 q_{1}^{1 / 2}+q_{2}$.

### a

Find Lucy's demand for $q_{1}$ and $q_{2}$ in terms of $Y, p_{1}$, and $p_{2}$.

### b
Lucy has income $Y=12$ and initially faces prices $p_{1}=2$ and $p_{2}=4$. What are her optimal quantities of $q_{1}$ and $q_{2}$ ?

### c
The price of $q_{1}$ increases to 4 . Find her new optimal quantities. What is the compensating variation?

## SP 22 Exam 1 Q1
Answer each of the following questions as either true or false. For a statement to be “true”, it must always be true. If there is a case where the statement is not true (or if you need more information to answer for sure), answer “false”. You must justify each answer with an appropriate explanation or counterexample (which may include a relevant diagram).

### a
Suppose the price of a good decreases, and the income effect is negative (i.e., the income effect directs the consumer to consume less of the good). Then, the good cannot be a Giffen good.

### b
Janet is making a BLT (bacon, lettuce, and tomato) sandwich, and prefers each sandwich to have one slice of tomato $(q_t)$, one piece of lettuce $(q_l)$, and three slices of bacon $(q_b)$. True or false: Janet’s preferences can be represented by the utility function $U(q_t, q_l,, q_b) = \min\{3q_t, 3q_l, q_b\}$.

### c

Last month, the price of books was $p_b = 10$ and the price of magazines was $p_m = 5$, and Sebastian bought the bundle $(q_b, q_m) = (3, 4)$. This month, the price of magazines has risen to $p_m = 10$. Let CV denote the compensating variation for this price change, and $|CV|$ its absolute value. True or false: $|CV |<$ $ $20$. (Assume that Sebastian has “normal” preferences with strictly decreasing MRS, and that he has the same income in both months.)

### d

The Cookie Monster currently has $4$ glasses of milk and $3$ cookies, and his MRS is
$$
MRS^C=\frac{\sqrt{q_m}}{q_c}.
$$

The Milk Monster currently has $6$ glasses of milk and $3$ cookies, and his MRS is
$$
MRS^M = \frac{q_m}{6 q_c}.
$$
True or false: both the Cookie Monster and the Milk Monster will be willing to participate in a trade in which the Cookie Monster gives up cookies in exchange for milk. (For the purposes of interpreting the MRS, treat milk as good $1$, and cookies as good $2$).