S old. S new. Old p D. Old q. New q

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1 Proble Set 1: Solutions ECON 301: Interediate Microeconoics Prof. Marek Weretka Proble 1 (Fro Varian Chapter 1) In this proble, the supply curve shifts to the left as soe of the apartents are converted into condoinius. The deand curve, however, is not changed; the inner ring people who were already renting apartents are still in the arket for apartents (they did not ove into the condoiniu arket only the outer-ring people are in that arket). The equilibriu rental price is higher, while the equilibriu quantity of apartent rentals in this arket is lower. Reservation Price, p S new S old New p Old p D New q Old q Quantity of Apartents, q Proble 2 (Murphy s Budget Set) (a) The forula for the budget constraint with two goods is x 1 + x 2. For this proble, = 2 (we ll just assue French Fries are x 1, since that s what we ll have on the horizontal axis), the price of Beef Jerky is = 5 per pack. With incoe = 100, Murphy s budget constraint is 2x 1 + 5x Notice that the budget constraint represents all the affordable (x 1, x 2 ) cobinations (anything less than or equal to $100). The budget line represents the (x 1, x 2 ) cobinations that cost exactly $100, so the equation representing the budget line is 2x 1 + 5x 2 = 100. (b) The real incoe in ters of French Fries (which is the portions of French Fries Murphy could consue if he spent all his incoe on French Fries) is = = 50. 1

2 Thinking of what this eans graphically, we just found the x 1 -intercept of budget line. (c) The real incoe in ters of Beef Jerky (which is the packs of Beef Jerky Murphy could consue if he were to consue only Beef Jerky) is = = 20. This tells us the x 2 -intercept of budget line. Notice that, accordingly, this is coing directly fro the budget line: If we set x 1 = 0 (no French Fries consued) in the budget line forula and solve for x 2, we have 0 + x 2 = = x 2 =, telling us how uch Beef Jerky can be consued if of French Fries are consued. (d) The red budget line (given by x 1 + x 2 = ) and the pink budget set area (all of the cobinations of x 1 and x 2 that are affordable x 1 + x 2.) are shown below. Notice that the quantities we found in (b) and (c) are indeed the x 1 - and x 2 -intercepts. Beef Jerky, x 2 = 20 slope = 2 5 = 50 French Fries, x 1 (e) The slope of the budget line is = 2. (Thinking way back to Algebra, just put the 5 budget line in slope-intercept for to see this.) The econoic interpretation is that this is the real price of French Fries in ters of Beef Jerky. I.e., to get one ore portion of French Fries, you would have to consue 2/5 fewer packs of Beef Jerky to afford that extra portion of Fries. (f) With the new, higher price of Beef Jerky, Murphy s total incoe in ters of Beef Jerky has fallen fro 20 packs to 10 packs (so the new x 2 -intercept is now 10). His real incoe in ters of Fries has not changed (therefore the x 1 -intercept is still 50). We can think of his budget line as rotating down as seen below. Notice that the new slope of the budget line (the relative price) is = 2 = 1. (While the actual price of Fries has not changed, 10 5 Fries are relatively cheaper copared to Beef Jerky than they were before.) 2

3 Beef Jerky, x 2 = 10 slope = 1 5 = 50 French Fries, x 1 (g) The Jerky Bill essentially prohibits Murphy fro consuing any cobination (x 1, x 2 ) cobination of French Fries and Beef Jerky with x 2 > 5. So the area of this budget constraint set that overlaps with the part of the graph where x 2 > 5 is off liits. Notice that this does not change the slope of the budget line exactly; it rather forces hi to choose aong only the affordable bundles for which x 2 5. Beef Jerky, x 2 5 = 50 French Fries, x 1 Proble 3 (Budget Set with Taxes) p (a) The relative price of a CD (x 1 ) in ters ilk (x 2 ) is: 1 = 20 = 20. So, along the 1 budget line, to purchase one ore CD, Ay ust forego 20 bottles of ilk. (Reeber that *analytically* the slope of the budget line still has that negative sign:. It s just that when we talk *econoically* about the relative price of soething like how uch we have to give up of one thing to get another the negative is iplied by the give up part.) 3

4 (b) Ay s budget set: Milk, x 2 =2, 000 slope = 20 = 100 CDs, x 1 (c) An ad valore tax here on the CDs is equivalent to a change in the price of CDs to p T 1 = (1 + tax), so p T 1 = (1 + 1) 20 = 40. The new relative price of a CD in ters of bottles of ilk is pt 1 p2 = 40 = 40. Her new budget line and budget set are shown graphically below: 1 Milk, x 2 =2, 000 slope = 40 p T 1 = 50 CDs, x 1 (d) If Ay is sent 5 CDs, then her new budget set includes the budget set shown in (c) plus up to 5 ore CDs at any point, shifting the budget line to the right by 5 units. The axiu nuber of CDs she can consue has increased by 5 everywhere, including the x 1 -intercept, which is now 55 instead of 50 (/p T = 55). The x 2 -intercept, however, has not changed, since she can still only purchase 2,000 bottles of ilk (i.e., we still have / = 2, 000). There is a kink in the budget line at (5, 2,000), which represents the bundle where 5 CDs (given to her) and 2,000 bottles of ilk (all of which she purchased using her 4

5 entire incoe). Milk, x 2 =2, 000 (5, 2000) p T = 55 CDs, x 1 Proble 4 (Indifference Curves) (a) If you like ice crea and chocolate a lot, then the indifference curves ust be downward sloping with utility increasing as we ove away fro the origin. (There s not enough inforation here to deterine whether the indifference curves are convex; we only know for sure that they have a negative slope.) Chocolate, x 2 Ice Crea, x 1 5

6 (b) If you like ice crea and hate chocolate, chocolate is a bad and the less you have of it, the better off you are. To see this, one way is to take any point in the consuption space and copare utility to the right, left, above, and below to deterine the direction of increasing utility. (Just like in (a), we can t tell whether the MRS is constant or not as we ove along an indifference curve, we only know that the slope is positive.) Chocolate, x 2 Ice Crea, x 1 (c) In this case, all that atters is ice crea, the ore the better; the aount of chocolate does not atter. Here, you are indifferent between points a and b, but you like c better than a (and b). Chocolate, x 2 b a c Ice Crea, x 1 (d) The sign of the MRS for the three preferences above (i.e., the slope of the indifference curves): (a): Negative. (You would give up soe chocolate to get ore ice crea to aintain the sae level of utility.) 6

7 (b): Positive. (You would have to get ore chocolate, which you dislike, in order to get ore ice crea and be indifferent to the change.) (c): Infinite. (Given a certain aount of ice crea, your utility level is the sae whether you were to consue infinitely less or infinitely ore chocolate.) (e) If you ust eat ice crea and chocolate in the sae proportion (i.e., the two goods are perfect copleents), then oving away fro the origin along the dotted line where this is true (where x 1 = x 2 ), you are better off. However, if we ove off the dotted line, say fro point a to b, or fro a to c, we are not any better off; we re just indifferent. Chocolate, x 2 x 1 = x 2 b a c Ice Crea, x 1 Proble 5 (Convexity and Monotonicity) (a) The preferences depicted in Figure 3.4 in Varian are onotone (but not strictly onotone) and convex (but not strictly convex). They are onotone because having ore of left and right shoes will increase utility; however the preferences are not strictly onotone since, for exaple, holding the nuber of right shoes fixed, getting ore left shoes does not ake one better off (it siply does not ake hi worse off). 7

8 (b) The preferences in Figure 3.7 are not onotone but they are strictly convex. To see that they are not onotone, copare points a and b below. Bundle b has ore of both goods x 1 and x 2, however a is preferred to b, so it s not always true that ore is preferred to less with these preferences. (Note that there are soe other points in the space where ore is preferred to less, but we only need to show one point where that s not the case to show that the preferences are not onotonic). To see that these preferences are strictly convex, consider points a and c. Any bundle that is a linear cobination of the two (along the dotted line) is strictly preferred to both a and c. Since this is true everywhere (i.e., we won t be able to find any point where this is not the case), these preferences are strictly convex. x 2 c a b x 1 8