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Dave is a 3rd grade student trying to decide what to have for lunch

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Dave is a 3rd grade student trying to decide what to have for lunch. He likes only Twinkies (x) and soda (y), and these provide him utility of u (x, y) = (x^2/3) * (y^1/3).

a. Write down the constrained utility-maximization problem for Dave, including (i) the objective, (ii) the objective function, (iii) the choice variables, and (iv) the constraint (note this is not the same as writing down the Lagrangian).

b. Write down the Lagrangian function for Dave’s constrained utility maximization problem and solve for the first-order conditions of the Lagrangian.

c. Solve for Dave’s Marshallian demands for x and y as functions of prices (px and py) and income (I), x* (px, py, I) and y* (px, py, I).

d. If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Dave spend the $3 his mother gives him to maximize his utility? How much does his optimal demand for soda (y*) change when the price of Twinkies (px) increases?

e. Show mathematically the change in x* that results from an increase in income (∂x*/∂I) and from an increase in the price of x (∂x*/∂px).