UCLA Economics 11 – Fall 2011 Professor Mazzocco MIDTERM 1, Version 1 NAME: _______________________________________ID:____________________ TA:__________________________________________________________________ Part I: Multiple Choice Questions (3 points each): 1. What is the x that minimizes the function22)ln( xxxy , where x>0? a. x=1/2 b. x= 2/1e c. none of the above d. does not exist 2. Which of the following marginal rates of substitutions is not decreasing? a. b. c. d. 3. When making her morning coffee, Julie prefers to add 1.5 packets of sugar over just 1 packet of sugar. She also prefers 1 packet of sugar to 2 packets of sugar and 2 packets of sugar to 1.5 packets of sugar. Which axiom of rational choice is violated by Julie’s preferences? a. Continuity b. Transitivity c. Ordinality d. Completeness4. When Germaine goes to Chipotle, he can never decide between 3 tacos (T) or 1 burrito (B) because he likes them equally. Which utility function could Germaine use to find his optimal consumption of burritos and tacos? a. U=2T+6B b. U=Min{1/3T,B} c. U=T1/3*B d. U=3T+B 5. Suppose an individual's MRS (of burgers for pizza) is 3:5. That is, at the current consumption choices he or she is willing to give up 5 burgers to get an extra 3 pizzas. Suppose also that the price of a burger is $5 and a pizza is $10. Then in order to increase utility the individual should a. buy more burgers and less pizza. b. buy more pizza and less burgers. c. continue with current consumption plans. d. none of the above 6. If an individual's utility function for salt and pepper is given by , the demand function for salt is given by a. b. c. d. 7. An individual has a utility function for skis (x) and ski boots (y) of the form U(x, y) = min(2x,y). His or her indirect utility function is given by: a.  yxppIV b.  IppVyx2 c.  IppVyx 2 d.  yxppIV228. Bob spends his income buying books (B) and music cds (M). The price of each book is 20 dollars; the price of each cd is 10 dollars. His utility is given by: U=lnB+lnM. What is the minimum income Bob needs to obtain a utility level equal to ln(8)? a, 40 b, 60 c, 80 d, 100 9. If utility is given by U(X, Y) = X+3Y, the price for X and Y are 1 and 2 respectively, the total income you have is 10. Which of the following bundle is your best choice a. (0, 5) b. (10, 0) c. (6, 2) d. (4, 4) 10. There are two types of goods, X and Y. The price of X is 2 and the price of Y is 1. At a certain bundle on the budget line, the MRS of X for Y is 3 ( ), what can you do to increase your utility? a. buy less X and less Y b. buy more X and less Y c. buy less X and more Y d. buy more X and more YPart II: Essay Questions Question 1 (35 Points) Mary’s Utility function is given by U(X,Y)=XaY1-a, the price of X and Y are given by px and py, respectively, and total income is equal to I. a) Write down Mary’s budget constraint. (5 points) pxX+ pyY=I For the rest of the question, consider the case in which a = 0.5. b) Find the consumption bundle Lisa will optimally choose. (7 points) X*=a*I/px, Y*=(1-a)*I/py c) Calculate Mary’s total expenditure on good X. Compute the share of income spent on the good X? (7 points) Total expenditure equals a*I, the share is a. d) Suppose now that the price of X doubles. Recalculate the share of income spent on the good X by Mary. What is the interpretation of the parameter a? (9 points) The share is a again. It represents the share Lisa will spend on X. e) Now suppose that Mary’s utility function is U(X,Y)=Xa+0.3Y1-a-0.3 and that the price of X goes back to px. Without solving the maximization problem, how many unit of X will Lisa buy now? (7 points) X=(a+0.3)*I/pxQuestion 2 (35 Points) John views goods X and Y as perfect substitutes in the proportion of X to of Y. His utility function is thus given by . a) For a given amount of income , and prices and , find John’s Marshallian demand functions for X and Y (You will have three cases depending on the price ratio). (5 points) Marshallian demand: 1) If : x and 2) I f : x and 3) If : X and Y any non-negative value as long as . b) Find the indirect utility function (You will have three cases depending on the price ratio). (5 points) 1) If : 2) If : 3) If : c) Suppose John has dollars to spend on X and Y, and the prices of these goods are and . What is his optimal consumption bundle and utility level? (5 points) x and The government wishes to raise revenues through taxes. Suppose first that it decides to impose an income tax of dollars on John (20 dollars are subtracted from Joe’s income). d) Find the optimal consumption bundle with the income tax. Find John’s new utility level. (6 points) x ande) Now suppose that instead of taxing income, the government places a tax of 10 dollars on the price of the good (a tax is a negative subsidy on the price of the good ; therefore follow what we did in class with the price subsidy, but instead of subtracting τ as we did in class add τ to the price) What is John’s new consumption and utility level? How much revenue does the tax raise? (6 points) New Px=20 x and f) Suppose the government decides to choose a tax on the price of the good that enables it to collect from John the same amount of tax revenues as the amount collected when it was imposing the income tax of 20 dollars. What is the tax on the price of the good that the government will choose? (8 points) We want to find τ such that: Revenue = (τ)(x*) = 20 Remember that since the tax is imposed on x, we need the agents to buy a positive quantity of x, therefore, we need: Which means that , as long as this is satisfied, then the optimal quantity of x is: x* Using the revenue restriction: (τ)( ) = 20  100τ = 200 + 20τ  which satisfies the restriction found