ANSWER KEY 1 BUDGETS


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1 ANSWER KEY 1 BUDGETS W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE (1) Draw the budget set for the following paraeters. =, p beer = 1, p pizza =. The units for the two goods are pints and pies respectively. (pints) MRT = pints beer per pizza (1/ pizza per pint) Beer Figure 1. ANSWER. The consuer s budget set for pizzabeer consuption bundles. Note that key features are labeled: the axes with the nae of their respective good and units, intercepts, the slope of the line (the MRT). The budget set itself is indicated by the shaded region. Note that the decisiont to put beer on the vertical axis and pizza on the horizontal is arbitrary. There is nothing especially horizontal about pizza or beer. It does not atter which way you choose as long as the picture is welllabeled. Note further that scale for each axis need not be the sae. You ay represent one pint of beer as an inch and 1 pizza as a eter or vice versa as long as you are consistent. The slope of the line as visually represented has no eaning without labels indicating units. Siilarly the nueric value reported for the MRT is unit dependent. Here we have 1 pizza pie worth 1 pint of beer at the given prices. But if we had used L as the unit for beer, we would report an MRT of 80: 1 pie per 80L of beer. (a) Fro any interior ( = not intercept point) consuption bundle on the budget line how any beers ust be forgone for each additional pizza? ANSWER. beers per pizza as indicated by the slope of the budget line in Figure 1. (b) How any pizzas ust be given up for each additional beer? ANSWER. 1 pizzas per beer as indicated by the inverse slope of the budget line in Figure 1. (c) Show in a new picture how the budget set changes when the price of beer increases to 2. ANSWER. See Figure 2 (d) Show in a new picture how the budget set changes when this consuer s incoe decreases to 1 ANSWER. See Figure 2 1
2 2 W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE Beer (pints) 10 Lost Choices Due to Price Increase MRT = 2. pints beer per pizza (2/ pizza per pint) Line Pivots Around Intercept on Axis of good whose price did NOT change. Beer (pints) 1 MRT = pints beer per pizza (1/ pizza per pint) (unchanged) New Line is parallel to old  shifted inward. Old Line Figure 2. The diagra on the left show the change in the budget set if the price of beer doubles fro $1 to $2 per pint. The diagra on the right shows the change in the budget set if incoe descreases fro $ to $1. (2) Suppose that there is unifor inflation by which I ean all prices increase by the sae rate r, so that the new prices (p 1,p 2) after the inflation are given by p 1 = p 1 (1 + r) and p 2 = p 2 (1 + r). Show how the budget set changes. If you wanted to restore the consuers budget set to what it was before the inflation without changes the prices, what could you do? ANSWER. the new level of incoe equal to = (1 + r). That is raise it by the sae rate as the prices increased. This will restore the budget set to exactly the sae set of consuption bundles as before the inflation. Note that a (2good) budget set can be described by paraeters: the two prices and the incoe. In Figure, the budget lines are labeled by the paraeter triples they are eant to represent. We exhibited a special property of budget sets: ultiplying all the paraeters that describe the set by the sae factor ((1 + r) in this case) preserves the eleents of the set exactly as they were. This property is called hoogeneity of degree zero. p 2 = p 2 p 2 MRT = p1 p 2 Good 2 Line (, p 1, p 2) Line given by (, p 1, p 2 ) AND (, p 1, p 2) p 1 p 1 = p 1 Good 1 Figure. The outer budget line represents the frontier of the set of choices before the inflation event. The inner budget line afterwards. Notes that the shift is parallel exactly as though incoe decreased. The slope of a budget line is deterined by the ratio of the price not their absolute agnitudes; Since both prices increased at the sae rate, their ratio and the slope of the budget line reained constant. () Malco eats roast beef and sokes Caels. Roast beef is $ per pound and Caels are $ per pack. He currently chooses to consue pounds of beef and packs of cigarettes per week.
3 ANSWER KEY 1 BUDGETS Caels (packs) Lost Choices Orginal Line with MRT = packs per pound beef New Line with MRT = packs per pound beef Gained Choices Roast Beef (lbs) Figure. Malco s for Beef and Cigarettes before and after the change in the the price of beef and incoe. (a) Draw Malco s budget set. What is Malco s weekly incoe?answer. Assuing Malco s preferences are onotonic (so that he ust choose a point on his budget line), his weekly incoe ust be + or $. (b) If Malco wanted to soke ore cigarettes at what rate would have to give up beef? AN SWER. Fro Figure, we can see that Malco ust give up packs of cigarettes per pound of beef. Therefore the answer to the inverse question is the inverse rate : pounds per pack. (c) Suppose that Malco s incoe increases to $ and the price of cigarettes siultaneously increases to $ per pack. Draw his new budget set. Not knowing anything about Malco s preferences except what was revealed by his oringinal choice, is it possible to say whether Malco is better off or worse off with the new budget set? Explain. ANSWER. In general, it ay be difficult to say whether or not a consuer is better or worse off after such a change in which soe choices are lost and others are gained. However we can use a little logic to ake a ore precise judgeent in this case. As we see fro Figure, the new budget line goes exactly through Malco s orginal choice on the old budget line. In other words, Malco can still afford to keep doing exactly what he was doing before. Therefore the budget line change could not have ade hi worse off. That is all we can say with certainty. However there is a good chance that he ay be strictly happier. We will exaine this situation in detail when we discuss incoe and substitution effects. () Consider a set of prices and incoe given by (p 1,p 2,) and fix a consuption bundle (x 1,x 2 ) on the resulting budget line. Suppose that the price of good one increases fro p 1 to p 1. Write down expression for a new level of incoe so that the resulting new budget line goes through (x 1,x 2 ). (Hint this is what occured in the previous proble.) ANSWER. = p 1x 1 + p 2 x 2. () Bulk Pricing Suppose a consuer has an incoe of $100 to spend on chocolate bars and icecrea cones. She faces a constant price of $1 for chocolate bars. However, the price of ice crea depends on how uch you buy. The first 10 ice crea cones cost the consuer $2 each, but every cone after that is $0.0 each. Draw the budget set. ANSWER see Figure.
4 W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE 100 MRT = 2 Bars per Cone Chocolate Bars 80 MRT = 1 2 Bar per Cone Ice Crea Cones Figure. for Ice Crea and Chocolate. Note that bulk pricing discount generates a inwardkinked budget line. In ters of Chocolate, the price of icecrea as indicated by the slope of the budget line changes fro 2 bar to 1 bar after the 2 10th cone. (6) Food Staps. Willia has $ to spend on food and a coposite good. The price of food is $1 per unit. The price of the coposite good is $1 per unit. (a) Draw Willia s budget set. ANSWER. The shaded region labeled (a) in Figure 6. (b) Suppose that the governent gives Willia $100 in food staps and strictly enforces a rule which prohibits Will fro using his food staps for any purchases other than food. Draw the new budget set. Describe the MRT of food for other goods along the budget line. ANSWER. The shaded regions labeled (a) and (b) in Figure 6. The arginal rate of transforation for the first $100 in food is $0 in other consuption. This is due to the fact that Willia cannot convert food stapts into anything but food. Therefore is he is at a point where he would be consuing less tahn $100 in food, it ust correspond to a situation where either he is not spending all of his food staps or he is throwing away food. Therefore the sacrifice in ters of other consuption to increase his food consuption is nothing. (c) Suppose that the rule prohibiting nonfood purchases is difficult to enforce and Willia can sell his food stap for $0.0 on the dollar. Draw his budget set. ANSWER. The cobined shaded regions labeled (a), (b) and (c) in Figure 6. Now along the region of his budget line where he consues between 0 and $100 in food his MRT is 1 2 dollar in other consuption per dollar of food.
5 ANSWER KEY 1 BUDGETS Other Consuption ($) 00 (c) (a) MRT = 1 2 $ in other consuption per $1 in food MRT = $0 in other consuption per $1 in food (b) MRT = $1 in other consuption per $1 in food 0 Food ($) Figure 6. Willia s budget for food and other consuption with (c) and without (b) a secondary arket for food staps. (7) Additional Practice Fro Workbook (Optional) 2.2, 2., 2.8, 2.11, 2.12
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