Log in
HomeClass Notes1,200,000US530,000

ECON 20000 Lecture Notes - Lecture 4: Marshallian Demand Function, Concave Function, Budget Constraint

113 views7 pages
salmonimpala389
8 Dec 2018
School
University of Chicago
Department
Economics
Course
ECON 20000
Professor
All

For unlimited access to Class Notes, a Class+ subscription is required.

Document Summary

The consumer chooses the bundle that maximizes utility subject to the budget constraint. U (x, y) max x,y s. t. pxx + pyy = m x 0 y 0. Suppose u is well-behaved: it is strictly increasing in both variables, it is continuously differentiable and strictly concave. Standard calcus tools for constrained-optimization: lagrangian of the utility maximization problem is the following three variable function: Using the de nintion of the lagrangian, the rst-order conditions are: u. Px = 0 which can be rewritten to: ux = px. Py = 0 which can be rewritten to: uy = py: pxx + pyy m = 0. Rearranging the rst order conditions give intuition: we see that is the marginal utility of a dollar. The utility-per-unit (marginal utility) divided by the dollars-per-unit (price) gives the utility-per-dollar. The marginal rate of substitution equals the negative of the price ratio.

Get access

Grade+
$40 USD/m
Billed monthly
Grade+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
10 Verified Answers
Class+
$30 USD/m
Billed monthly
Class+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
7 Verified Answers

Related Documents

ECON 20000 Lecture Notes - Lecture 6: Hicksian Demand Function, University Of Houston, Expenditure Function
salmonimpala389
ECON 20000 Lecture Notes - Lecture 8: Envelope Theorem, Expenditure Function, Inferior Good
salmonimpala389
ECON 101A
Final Exam
Study Guide
ECON 101A Study Guide - Final Guide: Utility Maximization Problem, Budget Constraint, Utility
turquoisegnu128

Related Questions

A consumer with the utility function U(X, Y) = [X – 10] Y^(1/2) with an income of $100 to spend on good X (with price $1) and good Y (with price $15).

1 Express this consumer’s utility maximization problem, using the particulars of this problem.

2 Give two conditions that the consumer’s optimal bundle will satisfy, again using the particulars of this problem.

3 Find the two conditions for the consumer’s optimal bundle. Neatly show your work.

4 Find the demand function for good X for a consumer with the given utility function. In other words, resolve this problem only now leaving the prices and income as variables: pX, pY, and m. Neatly show your work.

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!)

Consider the one period model with endogenous labor supply we studied in class, where workers (consumers) are born with one unit of labor effort and no physical endowments. The total population of consumers is one. Furthermore, assume that workers have the following lifetime utility function: U(c, l) = ln c + ln l Production takes place by a representative firm that has access to a production technology of the Cobb-Douglas form: Y = AK α L 1−α and as usual profits are rebated back to consumers and P = 1. Unlike the model discussed in class, the firm pays (w + τ ) for every unit of labor time used, τ > 0 is a unit tax imposed by the government (or some type of regulation). In this manner, the firm pays τL to the government and wL to workers. Finally, the government does no collect taxes from workers but only from employers as mentioned above. Government revenue is used to finance spending, G where G does not directly affect the economy. The government’s budget constraint is such that: τL = G

A. Write down the individual’s problem of utility maximization. Be sure to derive the budget constraint for the problem. Provide economic interpretation for this model.