A utility-maximizing consumer has the following utility function: \( U(X,Y) = 2X^{1/2} + Y \).

a. Are these preferences strictly monotonic? Strictly convex? Explain.

i. The preferences represented by this utility function are:
- A. Strictly monotonic; weakly convex
- B. Weakly monotonic; weakly convex
- C. Weakly monotonic; strictly convex
- D. Strictly monotonic; strictly convex

b. Assuming our utility maximizer faces a linear budget constraint of the form \( P_X X + P_Y Y = I \), derive his/her optimal demands \( X^*(P_X, P_Y, I) \) and \( Y^*(P_X, P_Y, I) \). You may use any method you prefer, but please show your work. If these demands are piecewise functions, then state each piece clearly (namely, is a corner solution possible, and, if so, what would \( X^* \) and \( Y^* \) be then?)

i. If \( I > 0 \), then this consumer's optimal demand for \( X \) is and his optimal demand for \( Y \) is:
- A. \( (P_Y / P_X)^2; I / P_X; 0 \)
- B. \( P_X / P_Y; P_Y / P_X; I / P_Y \)
- C. \( P_X / P_Y; I / P_X; 0 \)
- D. \( (P_Y)^2 / P_X; (P_Y / P_X)^2; (1/P_Y) - (P_Y / P_X) \)

ii. If \( I \) is less than or equal to a certain value, then the optimum is a corner solution, and he buys:
- A. \( (P_Y)^2 / P_X \); all \( Y \) and no \( X \).
- B. \( (P_Y / P_X)^2 \); all \( Y \) and no \( X \).
- C. \( (P_Y)^2 / P_X \); all \( X \) and no \( Y \).
- D. \( P_X / P_Y \); all \( X \) and no \( Y \).

c. Assuming an interior optimum, what is the own-price elasticity of demand for \( X \)? The cross-price elasticity of demand for \( X \) with respect to \( P_Y \)?

i. At any interior optimum here, the own-price elasticity of demand for \( X \) is:
- A. -1
- B. -2
- C. 0
- D. -1/2

ii. At any interior optimum here, the cross-price elasticity of the demand for \( X \) with respect to \( P_Y \) is, implying that \( X \) and \( Y \) are:
- A. 2; substitutes
- B. 1; complements
- C. 2; complements
- D. 1; substitutes

d. Graph the Engel curve for \( Y \). Label intercept and slope values carefully.

i. Which of the following graphs (located in the PDF at the end of these questions) correctly depicts the Engel curve for \( Y \)?
- A. A
- B. D
- C. C
- D. B

e. Suppose, initially, that \( I = 100 \), \( P_X = 2 \) and \( P_Y = 2 \). Graph this consumer's initial optimum point (A) in a well-labeled graph. Your indifference curve should show whether strict convexity holds and whether it can hit an axis (or both).

i. At these initial price and income levels, this consumer maximizes utility by consuming units of \( X \) and units of \( Y \):
- A. 2; 49
- B. 1; 50
- C. 2; 50
- D. 1; 49

The price of \( X \) falls to 1. Find the consumer's new optimum point (C) and label it on the same graph as in (e). Draw another indifference curve to show this new optimum.

i. As a result of this fall in the price of \( X \), this consumer now maximizes utility by consuming units of \( X \) and units of \( Y \):
- A. 4; 48
- B. 2; 48
- C. 2; 46
- D. 4; 46

Find the coordinates of the substitution effect point (B) of this price change. Label it and draw the compensated budget line tangent to the original indifference curve through point A. [Hint: What do we know about good \( X \) for interior solutions here?]

i. The equation of this consumer's initial utility-maximizing indifference curve is:
- A. \( U_0 = 49 \) jollies
- B. \( U_0 = 48 \) jollies
- C. \( U_0 = 51 \) jollies
- D. \( U_0 = 50 \) jollies

ii. Because for interior solutions, \( X \) is a(n) good, point B lies in relation to point C:
- A. Normal; due South
- B. Normal; due West
- C. Neuter; due South
- D. Inferior; due West

iii. The coordinates of point B (substitution effect point) are:
- A. \( X_B = 1; Y_B = 48 \)
- B. \( X_B = 1; Y_B = 47 \)
- C. \( X_B = 4; Y_B = 47 \)
- D. \( X_B = 4; Y_B = 50 \)