1. Lagrange multiplier as shadow price of income. Consider a consumer withutility function u (x1; x2) = x1x2. She has income M and faces prices p1; p2:Using Lagrange method derive her optimal consumption bundle (x1; x2) andthe level of utility at the optimum u(x1; x2): This is a function of (p1; p2; M ) :Show that du(x1; x2)=dM = ; the value of the Lagrange multiplier.2. Exercise 2.2 from Lengwiler.3. Exercise 2.3 from Lengwiler.

1. (6 points) Suppose a consumer has a utility functionU(x 1​ ,x 2​ )=x 1100​ x 2100​ and faces a budget constraintp 1​ x 1​ +p 2​ x 2​ =M. Derive her demand functions for good 1 and good 2 . Verify that both functions are homogeneous of degree 0 in(p 1​ ,p 2​ ,M). 2. (6 points) Suppose a consumer has a utility functionU(x 1​ ,x 2​ )=(x 1​ +x 2​ ) 2 and faces a budget constraintp 1​ x 1​ +p 2​ x 2​ =M. Derive her demand functions for good 1 and good 2 .

U(x, y) = (3x + 2y)2 .The price of x is px = $10 per unit and his income is $200.(a) Obtain the equation of Johnathan’s indifference curve for the utility level U = 100. Drawthis indifference curve. (2 marks)(b) The price of y is py = $8 per unit. Obtain the marginal rate of substitution (MRS) and theequation of the budget line. Using a graph, find Johnathan’s optimal consumption bundle.In this graph, show the budget line, the optimal bundle, and the corresponding indifferencecurve. Make sure to label carefully all the curves. (3 marks)(c) Suppose that the price of y drops to py = $6 per unit (the price of x remains the same,at px = $10 per unit, and the income remains the same). Obtain the equation of the newbudget line. Using a new graph, find Johnathan’s optimal bundle with this new price. Inthis graph, show Johnathan’s new budget line, new optimal bundle, and the correspondingindifference curve. Make sure to label carefully all the curves. (3 marks)(d) Now suppose that the price of y is py = $8 per unit if Johnathan buys less than 10 units ofthis product, and py = $6 per unit if he buys 10 units of y or more (as an example, 20 unitsof y would cost $120). Assume that the price of x remains the same, at px = $10 per unit.Derive the equation of the budget line and draw it in a separate graph. (3 marks)(e) Using a new graph, find the optimal bundle(s) for the problem in part (d). In this new graph,show the budget line, the optimal bundle and corresponding indifference curve. Make sureto label carefully all the curves. (3 marks)

Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.

Consider a consumer whose preferences can be represented by a utilityfunction, U : R2+ −→ R, of the form U (x1, x2) = −x21 − x22. Suppose thatthis consumer’s budget set is given byB (p1, p2, y) = {(x1, x2) ∈ R2+ : p1x1 + p2x2 6 y} ,where (p1, p2, y) ∈ R3++ are arbitrary “prices and income parameters”.1. Find this consumer’s Marshallian demand function or correspondence.Justify your answer.2. Find this consumer’s indirect utility function. Justify your answer.

Collin likes milkshakes (m) and sushi (s). His preferenes over these two goods are representedby the following utility functionU (m, s) = 2√m + s.Collin’s income is $100 and the price of sushi is $10.(a) Suppose the price of milkshakes is initially $2. Find Collin’s optimal consumption bundle.