Question

1. Al Einstein has a utility function that we can describe by u(x₁, x₂) = x^2₁ + 2x₁x₂ + x^2₂
. Al’s wife, El Einstein, has a utility function v(x₁, x₂) = x₂ + x₁.
(a) Calculate Al’s marginal rate of substitution between x₁ and x₂.
(b) What is El’s marginal rate of substitution between x₁ and x₂?
(c) Do Al’s and El’s utility functions u(x₁, x₂) and v(x₁, x₂) represent the same preferences?
(d) Is El’s utility function a monotonic transformation of Al’s?

2. Marie Curie has a utility function U(x, y) = max{x, 2y}
(a) Draw a graph, and plot the lines given by the equations x = 10, and 2y = 10.
(b) What’s the value of U(x, y) when x = 10 and 2y < 10?
(c) What’s the value of U(x, y) when x < 10 and 2y = 10?
(d) On the same chart, plot the indifference curve along which U(x, y) = 10.
(e) Are Marie’s preferences convex?

3. Convert these marketing pitches into mathematical utility functions:
(a) Everything goes with Crackers
(b) Bank Brewery asks that you drink responsibly.
(c) Sunflower margarine tastes just like butter!

4. Suppose each individual in a society gets utility from varying degrees of a specific policy P, which can
be between 0 and 1 so that 0 < P < 1. Individual 1’s utility function is given by u₁(P) = P^0.5,
while individual 2’s utility function is given by u₂(P) = 0.3P, individual 3’s utility function is given by
u₃(P) = P^-0.5, and individual 4’s utility function is given by u₄ = u₁(P) + u₂(P) + u₃(P). Assuming
these are cardinal utility functions:
(a) In words, describe how the policy affects each individual.
(b) Calculate the utility of each individual for each value of P from 0.05 to 0.95 in increments of 0.5.
(c) There is no correct answer to this question: how would you choose a value of P if you were the
government? Why? What value of P would you choose?
(d) Why is there no correct answer to the question above?
(e) Which one of these individuals would you elect as your leader and why?

5. Here are some utility functions. Find a) the marginal utility with respect to
x₁, b) the marginal utility with respect to x₂, c) the marginal rate of substitution between x₁ and x₂.
(a) u(x₁, x₂) = 2x₁ + 3x₂
(b) u(x₁, x₂) = 4x₁ + 6x₂
(c) u(x₁, x₂) = ax₁ + bx₂
(d) u(x₁, x₂) = 2√x₁ + x₂
(e) u(x₁, x₂) = ln(x₁) + x₂
(f) u(x₁, x₂) = v(x₁) + x₂
(g) u(x₁, x₂) = x₁x₂
(h) u(x₁, x₂) = x^a₁x^b₂

(i) u(x₁, x₂) = (x₁ + 2)(x₂ + b)
(j) u(x₁, x₂) = (x₁ + a)(x₂ + b)
(k) u(x₁, x₂) = x^a₁ + x^a₂

6. Tom Edison is eating 5 avocados and 20 sugar apples. The marginal rate
of substitution is the rate at which Tom is willing to trade sugar apples for an extra avocado. At his
current consumption bundle of 5 avocados and 20 sugar apples, his marginal rate of substitution is 4.
Compared to his current
(a) Will Tom be better off if he instead consumes 6 avocados and 20 sugar apples?
(b) Will Tom be better off if he instead consumes 6 avocados and 15 sugar apples?
(c) Will Tom be better off if he instead consumes 7 avocados and 16 sugar apples?
(d) Will Tom be better off if he instead consumes 6 avocados and 16 sugar apples?
7. (To be done on your own) Give three examples of utility functions that give exactly the same preferences
as u(x, y) = xy. Derive the marginal rate of substitution in each case and check whether they are equal.

Answer 1

1

Sol

Given , Al's utility function u(x₁ , x₂ ) = x₁² + 2x₁x₂ + x₂²

Al's wife's utility function u(x₁ , x₂) = x₁ + x₂