(b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300?


 Roland Mills
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1 Micro PS2  Choice, Demand and Consumer Surplus 1. We begin again with Charlie of the apples and bananas. Recall that Charlie s utility function is U(x A, x B ) = x A x B. Suppose that the price of apples is 1, the price of bananas is 2, and Charlie s income is 40. (a) On the graph below, draw Charlie s budget line. (Use the graphs marks and try to make this line accurate putting bananas on the yaxis.) Plot a few points on the indifference curve that gives Charlie a utility of 150 and sketch this curve. Now plot a few points on the indifference curve that gives Charlie a utility of 300 and sketch this curve. (b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300? (d) On your graph, mark a point that Charlie can just afford (ie. on the budget line) and that gives him a higher utility than 150. Label that point A. (e) Neither of the indifference curves that you drew is tangent to Charlie s budget line. Let s try to find one that is. At any point, (x A, x B ), Charlie s marginal rate of substitution is a function of x A and x B. In fact, if you calculate the ratio of marginal utilities for Charlie s utility function, you will find that Charlie s marginal rate of substitution is MRS(x A, x B ) = x B /x A. This is the slope of his indifference curve at any (x A, x B ). The slope of Charlie s budget line is what? (Give an actual number): 1
2 (f) Write an equation that says that the budget line is tangent to an indifference curve at (x A, x B ): There are many solutions to this equation. Each of these solutions corresponds to a point on a different indifference curve. Draw a line that passes through all of these points. (g) The best bundle that Charlie can afford must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can t afford it. If the point lies inside of his budget line, he can afford to do better by buying more of both goods. On your graph, label this best affordable bundle with an E. This happens where x A =? and x B =?. Verify your answer by solving the two simultaneous equations given by his (i) the budget equation and the (ii) tangency condition (MRS = ratio of prices). (HINT: 1st get both equations. 2nd solve one for either x A or x B. 3rd plug this into the other equation and solve for the unknown. 4th use this answer to plug back into the budget equation and solve for the other.) (h) What is Charlie s utility if he consumes the bundle (20, 10)? (i) On the graph above, use a dotted line to draw his indifference curve through (20, 10). Does this indifference curve cross Charlie s budget line, just touch it, or never touch it? 2. Charlie consumes apples (x a ) and bananas (x b ). His budget line is p a x a + p b x b = m. The slope of his indifference curve at a bundle (x a, x b ) is: MU 1(x a,x b ) MU 2 (x a,x b ) = x b/x a. The slope of his budget line is p a /p b. a. Charlies indifference curve will be tangent to his budget line at the point (x a, x b ) if the follow equation is satisfied: b. You have two equations, the budget line and the tangency equation, that must be satisfied by the optimal bundle. Solve these two equations for x a and x b. Charlies demand for apples is x a (p a, p b, m) = His demand for bananas is x b (p a, p b, m) =. c. In general, the demand for both commodities will depend on the price of each and on income. But for Charlie the demand only depends on income and the price of that good. He always spends 2
3 the same fraction of his income on bananas. What is that fraction? 3. Linus has the utility function U(x, y) = x + 3y. (a) On the graph below, draw the indifference curve passing through the point (x, y) = (3, 3). Now sketch the indifference curve connecting bundles that give Linus a utility of 6. (b) On the same graph, use a dotted line to draw Linus s budget line if the price of x is 1 and the price of y is 2 and his income is 8. What bundle does Linus choose in this situation? (c) What bundle would Linus choose if the price of x is 1, the price of y is 4, and his income is 8? 4. (Here you will walk through a unique characteristics of quasilinear preferences  see pg 62 and 102 for brief discussion) Donald only consumes stamps and Twinkies. His preferences take the form u(s, t) = s + ln(t). Stamps and Twinkies have prices p s and p t, and he has income m. a. Write an expression that says the ratio of Donald s marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio of the price of Twinkies to the price of stamps. (Hint: his marginal utility of Twinkies is 1/t, and his MU of stamps is 1.) b. You can use the equation you found in the last part to show that if he buys both goods, Donald s demand function for Twinkies depends only on the price ratio and not on his income. What is Donald s demand function for Twinkies? (Note: the equation found in (a) is the tangency point 3
4 needed for an interior solution) c. Notice that for this special utility function, if Donald buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables (m, p t, p s ), show this. (Hint: what is the amount spent on Twinkies?) d. Since there are only two goods, any money that is not spent on Twinkies must be spent on stamps. Use the budget equation and Donald s demand function for Twinkies to find an expression for the number of stamps he will buy. e. The expression you just wrote down is negative if m < p s. Surely it makes no sense for him to be demanding negative amounts of postage stamps. If m < p s, Donald s demand for postage stamps will be 0. What would his demand for Twinkies be? f. Donald s wife complains that whenever Donald gets an extra dollar, he always spends it all on stamps. Is she right? (Assume that m > p s.) g. Suppose that the price of Twinkies is $2 and the price of stamps is $1. Draw Donald s Engel curve for Twinkies and stamps  be sure to mark which is which. (Hint: First draw the Engel curves for incomes greater than $1, then draw them for incomes less than $1.) 5. Bob likes beef, his demand for kilograms of beef is: D(p) = 100 p, where p is the price of a kilo of beef. a. If the price of beef is 50, how much will he consume? 4
5 b. How much gross consumer s surplus does he get from his consumption? c. How much money does he spend on beef? d. What is his net consumer s surplus? 6. In the graph below, you see a representation of Sarah Gamp s indifference curves between cucumbers and other goods. Suppose that the reference price of cucumbers and the reference price of other goods are both 1. (a) What is the minimum amount of money that Sarah would need in order to purchase a bundle that is indifferent to A? (b) What is the minimum amount of money that Sarah would need in order to purchase a bundle that is indifferent to B? (c) Suppose that the reference price for cucumbers is 2 and the reference price for other goods is 1. How much money does she need in order to purchase a bundle that is indifferent to bundle A? (d) What is the minimum amount of money that Sarah would need to purchase a bundle that is indifferent to B using these new prices? 5
6 (e) No matter what prices Sarah faces, the amount of money she needs to purchase a bundle indifferent to A must be (higher or lower?) than the amount she needs to purchase a bundle indifferent to B. 6
servings of fries, income is exhausted and MU per dollar spent is the same for both goods, so this is the equilibrium.
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