Last week, neither Alistair nor Bodana purchased blueberries (x). Both consumers had $30 to spend on blueberries and/or raspberries (y), the price of the two fruits was $3 per pack and both consumers bought raspberries only. Alistair's utility for blueberries and raspberries is UA(x,y) = x+2y. Bodana's utility for blueberries and raspberries is UB(x,y) = x +12ln(y). a) In two separate graphs, illustrate Alistair's and Bodana's optimal choices. Your graphs must illustrate the consumers' budget constraints and their indifference curves at the optimal bundle. b) At their optimal bundle, what is Alistair's marginal rate of substitution of raspberries for blueberries (- MUx/MUy)? At their optimal bundle, what is Bodana's marginal rate of substitution? This week, blueberries cost $3 per pack just like last week, however raspberries now cost $4 per pack. c) How many packs of blueberries and raspberries will Alistair buy this week? Clearly justify your answer. d) Using the Lagrangian method, find how many packs of blueberries and raspberries will Bodana buy this week. a. Write the expression of the raspberry lover's Lagrangian function. b. Take the derivatives respect to x, respect to y, and respect to λ (first order conditions of the Lagrangian problem) and equate them to zero. C. Solve for the endogenous variables x, y, and 2. Do you find acceptable (non-negative) values for all variables? e) Now compute Bodana's marginal utility from an extra dollar spend on blueberries (MUx/Px) and her marginal utility from an extra dollar spent on raspberries (MUy/Py). Contrast and compare these values and the value you obtained for the Lagrangian multiplier in part d) c. f) What is the lowest price for raspberries that would convince Alistair to spend some money on blueberries?

Problem 1 Last week, neither Alistair nor Bodana purchased blueberries (x). Both consumers had $30 to spend on blueberries and/or raspberries (y), the price of the two fruits was $3 per pack and both consumers bought raspberries only. Alistair's utility for blueberries and raspberries is UA(x,y) = x+2y. Bodana's utility for blueberries and raspberries is UB(x,y) = x +12ln(y). a) In two separate graphs, illustrate Alistair's and Bodana's optimal choices. Your graphs must illustrate the consumers' budget constraints and their indifference curves at the optimal bundle. b) At their optimal bundle, what is Alistair's marginal rate of substitution of raspberries for blueberries (- MUx/MUy)? At their optimal bundle, what is Bodana's marginal rate of substitution? This week, blueberries cost $3 per pack just like last week, however raspberries now cost $4 per pack. c) How many packs of blueberries and raspberries will Alistair buy this week? Clearly justify your answer. d) Using the Lagrangian method, find how many packs of blueberries and raspberries will Bodana buy this week. a. Write the expression of the raspberry lover's Lagrangian function. b. Take the derivatives respect to x, respect to y, and respect to λ (first order conditions of the Lagrangian problem) and equate them to zero. C. Solve for the endogenous variables x, y, and 2. Do you find acceptable (non-negative) values for all variables? e) Now compute Bodana's marginal utility from an extra dollar spend on blueberries (MUx/Px) and her marginal utility from an extra dollar spent on raspberries (MUy/Py). Contrast and compare these values and the value you obtained for the Lagrangian multiplier in part d) c. f) What is the lowest price for raspberries that would convince Alistair to spend some money on blueberries?