Question

A consumer has utility function U(x,y)= xy/(2x +y)

(a) In the same graph, draw the indifference curves {(x,y)∣U(x,y)=2} and {(x,y)∣U(x,y)=4} and identify the corresponding upper contour sets.
(b) Are these preferences monotone?
(c) Are these preferences convex? The picture you drew in part (a) suggests the answer is yes, but you need to justify your answer formally. Hint: for a fixed α>0, consider the upper contour set {x∣U(x)≥α}. For any two bundles y and z in the upper contour set and any λ∈[0,1] you need to check that U(λz+(1−λ)y)≥α (d) For an arbitrary price vector p∈R+²
and income I, solve the consumer's optimization problem and construct the corresponding Marshallian demand and indirect utility functions.

Answer 1

To draw the indifference curves and identify the corresponding upper contour sets for the given utility function. To determine if the preferences are monotone and convex. To solve the consumer's optimization problem and find the Marshallian demand functions and indirect utility function.

Step-by-step explanation:

To draw the indifference curves and identify the corresponding upper contour sets for the utility function U(x,y)= xy/(2x +y), we need to find the values of x and y that satisfy U(x,y) = 2 and U(x,y) = 4. We can start by rearranging the utility function equations to solve for y in terms of x. For U(x,y) = 2, the equation becomes 2(2x + y) = xy, which simplifies to y = 4x/(2-x). Solving for y in terms of x for U(x,y) = 4 gives y = 8x/(2-x). Using these equations, we can plot the indifference curves on a graph.

To determine if the preferences are monotone, we need to check if the marginal rate of substitution (MRS) is always decreasing. The MRS is given by the partial derivative of the utility function with respect to y divided by the partial derivative with respect to x, i.e., MRS = (dU/dy)/(dU/dx). If the MRS is always decreasing, then the preferences are monotone.

To determine if the preferences are convex, we need to check if the utility function satisfies the property of boundedness in the upper contour sets. If for any x and y in the upper contour set and for any λ in the range [0,1], U(λz + (1-λ)y) ≥ α, where α is a fixed value, then the preferences are convex.

For the last part of the question, solving the consumer's optimization problem involves maximizing the utility function subject to a budget constraint. Given a price vector p ∈ R+² and an income I, we can set up the Lagrangian function and find the first-order conditions for maximization. Solving these first-order conditions will give us the Marshallian demand functions for x and y, as well as the indirect utility function.