ECON201 Lecture Notes - Lecture 7: Slutsky Equation, Budget Constraint, Utility Maximization Problem

Given prices for two goods PX = 2 and PY = 3 and a budget of $120 a consumer is trying to maximize the utility function U(X, Y ) = 2X0.3Y 0.7 . Use the Lagrange method to solve this problem, but don’t worry about the second order conditions (if you wish, you can actually check that the MRS is decreasing and this utility function is well behaved). a) Write and solve the utility maximization problem subject to the budget constraint. What is the resulting value of the utility function? (U=49.0481) b) Write and solve the dual problem. Do you get the same optimal solution (the same optimal combination of X and Y)? What is the resulting expenditure? Does that match the budget of your original (primal) problem? (X=18, Y=28 should be the same as for a or very close due to rounding) c) Assume now general values for prices and income. Solve for the optimal amounts of X and Y as functions of PX, PY , and I. What conclusion can you draw regarding how you spend your budget when dealing with Cobb Douglas functions with exponents that add up to 1. You can actually try to solve the general case where we replace the exponents 0.3 and 0.7 with some α and β, where α + β = 1. (Hint: you should find that you spend percent of your income on X and β percent of your income on Y. See if you can get there by solving!)

PROBLEM 1 (This one is designed to give you lots of math practice!)

Consider the following individual utility functions:

Adam Smith: U(x,y) = xy

Jeremy Bentham: U(x,y) = x^ay^b (Cobb-Douglas utility function)

Alfred Marshall: U(x,y) = ln x + ln y

John M. Keynes: U(x,y) = x + y^b (Quasi-linear utility function)

Joan Robinson: U(x,y) = aX + bY (linear; perfect substitutes)

a. Solve for the MRS(XY) for each of the gentlemen (i.e., all but Joan Robinson). Show that Mr. Smith, Mr. Bentham, and Mr. Marshall all have diminishing MRS(XY), but that if a=b=2, their preferences exhibit constant, increasing, and decreasing marginal utility respectively. (Fortunately, diminishing MRS is the condition that guarantees convex indifference curves, so we can solve for their optimal consumption!)

b. What is the general formula for Mrs. Robinsons (constant) MRS?

c. Find demand functions for x and y for Smith, Bentham, Marshall, and Keynes. Assume that each man has income I and that the prices of x and y are p_x and p_y respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, p_x = 10 and p_y = 20. Use these values to find the utility-maximizing quantities of x and y consumed by each man.

e. Find an expression for the share of income Mr. Bentham and Mr. Marshall each devote to the consumption of good x, where this share, s_x, is defined as

(p_x x)/I.

f. Find expressions for the income elasticity of demand for good x for Mr. Bentham and Mr. Marshall. Find expressions for the (own) price elasticity of demand for good y for these two men.

Need help explaining PART C THANK YOU!

Suppose that two individuals, Jon and David, form a community and would like to construct a communal fort that would protect them from attacks. They both consume good X, a private good, and the protection of the fort, P. One unit of good X costs 1 unit of currency, and one unit of P costs 2 units of currency. Both Jon and David have an income of 100 and a utility function of the form:


U = log(X_i) + 2 × log(P_J + P_D)
The budget constraint for each is given by:
X_i + 2 × P_i = 100


(a) Find the amount of protection Jon will provide as a function of how much David provides, and explain why the relationship is the way it is.


(b) How much protection P will be privately provided in this case?


(c) Explain the economic intuition behind this amount, and compare it to the socially optimal amount without solving for the socially optimal amount.

HERE IS WHAT I DID SO FAR

A_ First you need to find the max utility, with is the utility and the budget constraint

U = log(X_i) + 2 × log(P_J + P_D) -> log(100-2PJ)+2log(PJ+PD)

Next you will need to differentiate the logarithim to get PJ

- 2 2

----------- + ----------------- = 0

100-2PJ PJ + PD

which we cal solve to generate

3PJ = 100 - PD

This can also be applied to PD

3PD = 100 - PJ

Using the substitute method we can find that (PJ = X; PD = Y)

3x = 100 - Y -> Y = -3x + 100

3(-3x + 100) = 100 - x

-9x + 300 = 100 - x

-8x = -200

x = 25

Which means that Y = 25

PJ = 25, PD = 25

25 +25 = 50