ECON GA 1003 Microeconomics (MA)– Spring 2021 MIDTERM EXAM

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Microeconomics (MA)– Spring 2021

ECON GA 1003

MIDTERM EXAM

Question 1. 15 Points

A consumer has a preference relation “≽ " defined over a consumption set X ⊆ ?! represented by the discontinuous utility function U(x) = {x} where {x} is the smallest integer n such that x ≤ n.

a) Illustrate the upper-contour set of bundle x = 2.5. This is the set of bundles z such that z ≽ x.

(keep in mind that the preference relation is defined over a unidimensional set).

b) When is a preference relation convex? Show that this preference relation is convex.

c) When is a preference relation continuous? Show that this preference relation is not continuous.

Question 2. 35 Points

A consumer has linear preferences represented by the utility function U(x,y) = x + βy where β ∈ (0,1).

a) Does the consumer consider both X and Y normal goods? Does he consider either good a necessity? Clearly justify your answer.

Assume that Px = Py = $1.

b) Write the Lagrangian for this consumer’s Utility Maximization Problem

c) Write the Kuhn Tucker conditions.

d) Solving the Kuhn Tucker conditions find the consumer’s optimal bundle as a function of the parameter β.

e) In this case, are the Kuhn-Tucker conditions sufficient for an optimum? Justify your answer.

d) Using the Hicksian method, in an indifference curve diagram clearly illustrate the substitution and the income effect of a decline of the price of Y from Py = $1 to P’y < α. Highlight the two effects both for good X and for good Y.

Use different colors or line textures to distinguish between indifference curves and budget lines.

Question 3. 35 Points

Consider a consumer with preferences summarized by the utility function U = {x0.5 + βy0.5}2 .

a) Solve for the Hicksian compensated demand and show that the compensated demand for good x is

b) Show that the expenditure function is

c) Verify Shephard’s Lemma

d) List two properties of expenditure functions and verify that this expenditure function satisfies them.

e) Use the duality results to find this consumer’s indirect utility function.

Question 4. 20 Points

Part A)

a) In a diagram measuring l/L along the horizontal axis and c along the vertical axis, draw the production set of a firm with technology that satisfies the properties P1 – P3 and increasing returns to scale.

b) Suggest a transformation function that could be consistent with your graph.

Part B) Consider a firm that can use inputs z1 and z2 to produce an output q according to the production function:

q = min[αz1, z2]

a) Find the firm’s conditional factors demand z = z(w1, w2, q, α)

b) Find the expression of the firm’s cost function C = C(w1, w2, q, α)

c) Show that the expression you have found is homogenous of degree one in w1 and w2.

d) Show that the expression you have found in part b) is non decreasing in w1.

Bonus

In Question 2, find the compensating variation of the price decline.