ECMB02F -- Problem Set 2

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Thomas Nichols
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1 1 ECMB02F -- Problem Set 2 You should do the assigned problems as the material is covered in class. Note: Odd numbered questions from the text have answers in the back of the text. 1. NICHOLSON - Do problems 3.3, 3.5, Suppose that there are only two goods - food and housing. The diagram labelled 2.1 shows the consumption bundles chosen by a consumer with indifference curves I 1, I 2, I 3, and I 4, when faced with different combinations of income (I) and prices for the two goods (P F and P H ). The bundles are: Point A B C D E F G H J K Food Housing The diagram labelled 2.2 duplicates diagram 2.1, leaving out the indifference curves (this may show the points and lines more clearly). a) If P F = 2, P H = 2, derive a table showing all the points on the Engel Curve for food. Repeat if P F = 3, P H = 2. Sketch each Engel curve. For P F = 3, P H = 2, sketch the Engel curve for housing. b) If I = 360 and P F = 6, derive a table showing the demand schedule for housing, and sketch the demand curve. Now instead set P F = 6 and hold utility constant on the I 2 indifference curve. Derive a table showing the compensated demand curve for housing, and sketch the curve on the same diagram as the regular demand curve. How are the two curves related? c) If I = 120 and P H = 2, derive the demand schedule for food. d) Is either good ever inferior? Is either good always normal? Is either good ever a Giffen good? Explain your answers. e) When I = 300, P H = 4, suppose that P F rises from 4 to 6. Using the points on the diagram, divide the change in the consumption of food into an income and a substitution effect. Explain in words what is signified by each effect. How much is this consumer harmed by this price rise (that is, how much additional income would we have to give this consumer to compensate for the price rise?)? f) When I = 120, P F = 2, suppose that P H falls from 4 to 4/3. Using the points on the diagram, divide the change in the consumption of housing into an income and a substitution effect. By how much does the price reduction raise this consumer's real income (that is, how much income could we take away from this consumer to compensate for the price reduction?)? g) How would a doubling in the price of food change the indifference curves in the original diagram?

4 4 3. Suppose that there are only two goods - X and Y. The diagram labelled 3.3 shows three of a consumer's indifference curves. a) When I = 90 and P X = 3, derive a schedule for the demand curve of Y. b) When I = 50 and P Y = 2, suppose that P X falls from 10 to 2. Divide the change in the consumption of X into an income and a substitution effect. What is the effect of the fall in price on real income? 4. An individual consumes two commodities, X and Y, and has the following utility function: U = XY. The price of X is 1 and the price of Y is also 1. The consumer's income is 120. a) Compute equilibrium X, Y, and U. b) The government levies a 100% tax on the price of X. Compute X, Y, and U. What revenue is collected by the government? c) Now remove the tax in b and replace it with a lump sum tax large enough to reduce the consumer's utility to an amount equal to that in part b. How large is that lump sum tax? In what sense does the difference between the amount raised by the taxes in parts b and c provide a measure of the inefficiency of the sales tax? (this is called the excess burden of the tax). d) Another way to show the inefficiency of the sales tax is to set the lump sum tax equal to the revenue raised by the sales tax and demonstrate that this results in a higher level of utility for the consumer. Do this. e) Repeat parts a, b, c and d when the utility function changes to: U = X 2/3 Y 1/3 f) Given what you have learned about lump sum taxes being efficient, why don't governments use them? 5. This problem pushes forward using the same functions we used in the previous problem set in question 6. a) Suppose U(x,y) = x 3 y. Find the demand curve for x, if I = 60, P Y = 5. If P x = 3, determine the size of the income and substitution effects for a change in the price of X. b) Continue with the same utility curve U(x,y) = x 3 y. If I = 60, P Y = 5, P x = 3, determine the value of utility. Holding that utility constant, find the compensated demand curve for x. By taking the derivative of this demand curve, confirm that you have the correct value for the substitution effect in part a. c) Again, continue with U(x,y) = x 3 y. Confirm that the demand curve for x does not change when P Y changes. Do you find this result unusual? d) Suppose U(x,y) = x 1/2 + y 1/2. Find the demand curve for x if I = 60, P Y = 2. If P x = 3, determine the size of the income and substitution effects for a change in the price of X. e) Suppose U(x,y) = (x -1 + y -1 ) -1. Find the demand curve for x if I = 150, P Y = 9. If P x = 4, determine the size of the income and substitution effects for a change in the price of X.

Economics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/ (a) The equation of the indifference curve is given by,

Dirk Bergemann Department of Economics Yale University Economics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/12 1. (a) The equation of the indifference curve is given by, (x 1 + 2)