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Economics 2150A/B Chapter Notes - Chapter 4: Composite Good, Budget Constraint, Revealed Preference

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coralporpoise546
19 Apr 2014
School
Western University
Department
Economics
Course
Economics 2150A/B
Professor
Prof

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ECON 2150A/B Full Course Notes
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ECON 2150A/B Full Course Notes
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Document Summary

Think of shifting the budget constraint inward to lower income levels until the expenditure is just tangent to the given utility curve u=u*. *be able to draw and label the budget constraints in these examples. When analyzing public policy it is often easiest to think of the consumer consuming the good of interest and a composite good h with a price of 1 dollar. Composite good: represents expenditure on all other goods and has a price set at . 00. If the composite good is on the vertical axis, then the y-intercept of the budget constraint represents both the quantity of the composite good purchased and total income: coupons versus cash subsidies. Note: with the composite good y on the y axis: slope of the b. c. Joe mv housing is high (steep ic). Jill mv housing is low (flat ic) would be better off with cash subsidy. P a g e | 3: joining a club.

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Related Questions

1. Caty consumes only goods X and Y . Her utility function is: U(X; Y ) = min{X; Y}. We

are given that PX = 3; PY = 6, and Caty's income is 18.


Calculate Caty's optimal consumption bundle, (X, Y). (Hint: Since Caty's indif-

ference curves are not smooth and \curvy", we cannot use MRS = MRT to solve

for the optimal bundle. Draw a diagram to see where Caty's optimal bundle must be

on her IC. How do you characterize this bundle mathematically?)


2. John consumes only goods X and Y . His utility function is: U(X, Y ) = X + 2Y . We are

given that PX = 3; PY = 3, and John's income is 30.


(a) Calculate the slopes of John's budget constraint and his indierence curves, as viewed

with Y as the vertical axis and X as the horizontal axis


(b) Calculate John's optimal consumption bundle, (X, Y). (Hint: Since John's indif-

ference curves are not smooth and \curvy", we cannot use MRS = MRT to solve for

the optimal bundle. Draw a diagram to see where the John's optimal bundle must

be on his IC. How do you characterize this bundle mathematically?)



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!)

Suppose that there are two goods (X and Y). The price of X is $2 per unitu, and the price of Y is $1 per unit. There are two consumers ( A and B). The utility functions for the consumers are: for consumer A: U (X,Y)= X^.5Y^.5 and for consumer B: U(X,Y)=X^.8Y^.2

Consumer A has an income of $100, and Consumer B has an income of $300.

1. Derive the marginal utlities of goods X and Y for consumer A


2. Derive the marginal utilities of goods X and Y for consumer B


3. Set consumer A's MRS equal to the price ratio. Then use the budget constraint to solve for her optimal bundle


4. Set consumer B's MRS equal to the price ratio. Then use the budget contraint to solve for his optimal consumption bundle.


5. Calculate the MRS for each consumer at their optimal consumption bundles.


6. Suppose that there is another conusmer (let's call her C). You don't know anything about her utility function or her income. All you know is that she consumes both goods. What do you know about C's MRS at her optimal consumption bundle? Why?