University of Pittsburgh-Pittsburgh Campus - ECON 1100

1)Originally the consumer faces the budget line p₁x₁ + p₂x₂ = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write dow an equation for the new budget line in terms of the original prices and income.

2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant?

3. If the price of good 1 doubles and the price of good 2 triples does the budget line become flatter or steeper?

4. Suppose the government puts a tax of 15 cents a gallon on gasoline and then later decides to put a subsidy on gasoline at a rate of 7 cents a gallon. What is the next tax in this combination equivalent to?

6. Suppose that a budget equation is given by p₁x₁+p₂x₂ = m. The government decides to impose a lump sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the new budget line?
7. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least as well off? How can we answer this question without knowing anything about the consumer’s preferences?
8.  If we observe a consumer choosing (x₁,x₂) when (y₁,y₂) is available one time, are we justified in concluding (

   

9. Consider a group of people A,B,C and the relation ’is at least as tall as,’ as in ’A is at least as tall as B.’. Is this relation transitive?

10. What is your marginal rate of substitution of $1 bills for $5 bills?
11. What kind of preferences are represented by a utility function of the form u(x₁,x₂) = (x₁ + x₂)^1/2 What about the utility function v(x₁,x₂) = 13x₁ + 13x₂?

12. What kind of preferences are represented by a utility function of the form

transformation of u(x₁,x₂) ?

13. Consider the utility function u(x₁,x₂) = (x₁x₂)^1/2. What kind of preferences does it represent? Is the function

    

     a monotonic transformation of u(x₁,x₂)? Is the function

    

     a monotonic transformation of u(x₁,x₂) ?
14. Can you explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution?
15. Draw the consumer’s budget constraint. Suppose m = 24, p₂ = 2 and p₁ depends on whether the consumer buys more or less than 12 units of x₁:

(

p1 =     1          when   x 1 ≤ 12

2          when   x₁ > 12

Can he consume the bundle (16,4) ?

16. What is the marginal rate of substitution?

17. For each case draw some indifference curves of an individual with the following preferences. Let x₁ = coffee and x₂ = cookies.
    1. I only care about my consumption of coffee, more cookies won’t make me any happier
    2. I only care about my consumption of cookies, more coffee won’t make me any happier
    3. I always consume exactly two cookies for every cup of coffee I consume.
    4. I care about my consumption of coffee; I dislike cookies
    5. I always care about 10 times as much about my consumption of coffee to my consumption of cookies
    6. I like both goods, but I prefer an average amount of both goods to a lot of only one.

18. Calculate the marginal rate of substitution for the following utility functions:
    1. u(x,y) = 3x + y
    2. u(x,y) = 3x²y
    3. u(x,y) = x + ln y
    4. u(x,y) = 3 ln x + 5 ln y
    5. u(x,y) = x³y⁵ F. u(x,y) = x1/2y2/3

19. For the following utility functions, solve for, and plot, the indifference curve passing through the bundle (x,y) = (2,3)
    1. u(x,y) = x^1/2 + y B. u(x,y) = xy

1. 3. u(x,y) = x + y
   4. u(x,y) = min{x,2y}

20. Suppose a consumer’s preferences are represented by the utility function u(x,y) = x⁵y. In addition, his income is m = 18, the price of the x good is p_x = 1, and the price of the y good is p_y = 1.
    1. What is the budget equation?
    2. What is the MRS for the consumer
    3. What is the tangency condition
    4. Solve for this consumer’s optimal bundle, then illustrate the optimal choice bundle, the budget set, and the indifference curve passing through the optimal bundle.
    5. Now suppose that income increases to m = 21. Solve for the new optimal bundle.

21. Suppose a consumer’s preferences are represented by the utility function u(x,y) = x+3y. In addition, his income is m = 18, the price of the x good is p_x = 1, and the price of the y good is p_y = 1.
    1. What is the budget equation?
    2. What is the MRS for the consumer
    3. What type of preferences does this consumer have?
    4. Solve for this consumer’s optimal bundle, then illustrate the optimal choice bundle, the budget set, and the indifference curve passing through the optimal bundle.
    5. Now suppose that the price of the x good falls to

       

       . Solve for the new optimal bundle.
    6. Now suppose that the price of the x good falls to

       

       . Solve for the new optimal bundle.

22. Suppose a consumer’s preferences are represented by the utility function u(x,y) = min{x,2y}. In addition, his income is m = 20, the price of the x good is p_x = 5 , and the price of the y good is p_y = 1.
    1. What is the budget equation?
    2. What type of preferences does this consumer have?
    3. Solve for this consumer’s optimal bundle, then illustrate the optimal choice bundle, the budget set, and the indifference curve passing through the optimal bundle.