--- 5. If Mary’s utility function takes the form \[ U = U(x_1, x_2) = (x_1^2 + 2x_2)(x_1 + 1)^2 \] Where \( U \) is the total utility, and \( x_1 \) and \( x_2 \) are the quantities of two commodities. (a) Find the marginal utility function of each of the two commodities. (8 pts) (b) Find the value of the marginal utility of \( x_1 \) when 3 units of each commodity are consumed. (8 pts) (c) Does Mary prefer more \( x_1 \) or less \( x_1 \)? Explain why. (8 pts)
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The utility function is given by: \[ U = (x_1^2 + 2x_2)(x_1 + 1)^2 \] **Marginal Utility of \( x_1 \)**: We will use the product rule for differentiation. Let \( A = x_1^2 + 2x_2 \) and \( B = (x_1 + 1)^2 \). Using the product rule: \[ MU_{x_1} = \frac{\partial Show more…
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Consider the following utility functions: (a) u(x1,x2) = x1.x2 (b) u(x1,x2) = (x1)^2.(x2)^2 where x1 is the amount of good 1 and x2 is the amount of good 2. (1) For each of the utility functions, find the marginal utility (MU,) with respect to good 1. (2)For each of the utility functions, find the marginal utility (MUz) with respect to good 2. (3)For each of the utility functions, consider the following indifference curve: k = u(x1, x2) for some constant k. Find the marginal rate of substitution (MRS) of good 1 for good 2 at a point (x1, x2) on the indifference curve. (4) Show that each of the utility functions has a diminishing MRS.
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11) A consumer's preferences over two goods (x1,x2) are represented by the utility function u(x1,x2) = x1^(1/2) + x2^(1/2). The income he allocates to the consumption of these two goods is m > 0. The price of the two goods are p1 and p2, respectively. a) Determine the monotonicity and convexity of these preferences and explain your reasoning. Briefly define monotonicity and convexity. b) Calculate the marginal rate of substitution (MRS(x1,x2)) between the two goods for this consumer. For the bundle (x1,x2) = (1,4), interpret the MRS. c) For any p1,p2, and m, calculate the demand functions of x1 and x2 including the corner solutions if there are any. Clearly state which assumptions you used to achieve your solutions. If there is no corner solution, discuss why this is the case. d) Consider a price increase in x1 from p1 to p1'. Find expressions for the substitution and income effects on x1 as a function of p1,p2, and m, and determine their signs. What do the signs of the substitution and income effects and the sign of the aggregate demand change tell you about these goods? Discuss.
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