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Department
Economics
Course
ECO206Y1
Professor
Robert Brym
Semester
Fall

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C H A P T E R 0 Foundational Preliminaries: Answers to Within-Chapter-Exercises 0A Answers for Section A: Graphical Preliminaries Exercise 0A.1 Consider the set [0,1) which includes the point 0, all the points in between 0 and 1 but not the point 1. Is this set convex? What about the union of this set with the point 1? What about the union of this set with the point 1.1? Answer: The set [0,1) is convex. So is the union of this set with 1, denoted [0,1] (as illustrated in the chapter). But the union with 1.1 is not convex because any line segment connecting 1.1 with a point in [0,1) is not fully contained in the union of [0,1) with 1.1. Exercise 0A.2 Is the set of all rational numbers a convex set? What about the set of all non- integers? Or the set of all irrational numbers? Answer The set of all rational numbers is not convex because any line segment connecting two rational numbers contain irrational numbers that arenot in the set of rational numbers. Exercise0A.3 Describe points B, C, D and E in Graph2 as a pair of real numbers. Answer B =(8,2), C =(6,5), D = (4,5), and E =(7,6). Exercise0A.4 Which of the sets in Graph3 is/are not convex? Answer Only the set in (a) is convex. Exercise 0A.5 Can you use points A and B to arrive at the same value for the slope? What about the points D and F or the points D and C? Answer: From A to B, we have a positive rise of 20 and a negative run of 10 — giving us a slope of −20/10 = −2. From D to F, we have a positive rise of 2 and a Foundational Preliminaries: Answers to Within-Chapter-Exercises 2 negative run of 1 — giving us a slope of −2/1 =−2. From D toC, we have a positive rise of 4 and a negative run of 2 — giving us a slope of −4/2 =−2. Exercise0A.6 Is the shaded set in Graph4 a convex set? Answer : Yes. Exercise 0A.7 Suppose the blue line in Graph 4 had a kink in it. This kink could point “in- ward” (i.e. toward to origin) or “outward" (i.e. away from the origin). For which of these would the shaded area underneath the kinked line become a non-convex set? Answer: If the kink points “inward”, we get a graph like panel (b) in Graph 2.4 of the text. Connecting B and A in that graph gives us a line segment that lies fully above the set defined by the kinked lines — implying the set below is non-convex. If the kink points “outward”, we get a graph like panel (a) of Graph 2.4 in the text. The set that lies underneath is then convex. Exercise0A.8 Check to see that the other intercepts (at B and A) are correctly labeled based on equation (4). Answer : Setting z = 0 and y = 0, the equation gives us 4x = 40 which implies x = 10 — the intercept at A. Setting x = 0 and z = 0, the equation gives us 2y = 40 which implies y =20 — the intercept at B. Exercise0A.9 Is the plane in Graph5 a convex set? Answer: Yes. Exercise 0A.10 Given the equation (4) that describes the 3-dimensional plane, what is the equation that describes the magenta line segment which intersects with the plane in panel (a)? Answer: Atthatslice, z = 10. Setting z equalto10intheequation4x+2y+z =40, we get 4x +2y +10 = 40 or 4x +2y = 30. Re-writing this in terms of y, we get the equation y =15−2x, an equation with vertical intercept of 15 and slope of −2. Exercise 0A.11 Suppose I like eating steak and will eat more steak as my income goes up. Which way will my demand curve for steak shift as my income increases? Can you think of any goods for which my demand curve might shift in the other direction as my income in- creases? Answer: As income increases, the demand curve for steak will shift to the right (or “down”). For some goods, our consumption might decrease as our income in- creases. For instance, perhaps we buy less pasta as we get richer — implying our demand curve for pasta shifts to the left (or “up”) as income increases. 3 0A. Answers for Section A: Graphical Preliminaries Exercise 0A.12 Coffee and sugar are complements for me in the sense that I use sugar in my coffee. Can you guess which way my demand curve for coffee will shift as the price of sugar increases? Answer : When the pricefor sugar increases, I will buy less sugar as I slide up the demand curve for sugar. Since sugar and coffee are complements, I will also buy less coffee — implying that my demand curve for coffee shifts to the left (or “up”). Exercise 0A.13 Ice tea and coffee are substitutes for me in the sense that I like both of them but will only drink a certain total amount of liquids. Can you guess which way my demand curve for coffee will shift as the price of iced tea increases? Answer: As the price of iced tea increases, I slide up on the demand curve for iced tea, implying I will buy less iced tea. Since iced tea and coffee are substitutes for me, I will likely buy more coffee — implying my coffee consumption goes up. So my demand curve for coffee will shift to the right (or “down”). Exercise 0A.14 How would the supply curve for a firm shift if the general wage rate in the economy increases? Answer : The supply curve would shift to the left (or “up”). Exercise0A.15 Would you expect the supply curve for a firm that produces x to shift when the price of some other good y (that is not used in the production of x) increases? Answer : The increase in the price of y is unrelated to the production costs for x — and so the supply curve would not shift. Exercise 0A.16 If the supply curve depicts the supply curve for a market composed of many firms, we may also see shifts in the supply curve that arise from the entry of new firms or the exit of existing firms. How would the market supply curve shift as firms enter and exit? Answer: As firms enter, the market supply curve would shift to the right (or “down”), and as firms exit, the market supply curve would shift to the left (or “up”). Exercise 0A.17 Is consumer 2’s consumption also more elastic than consumer 1’s when price falls? Answer: Yes, when price falls, the green consumer 2 will increase consumption more than the magenta consumer 1. Exercise0A.18 Do slopes similarlychange if we measure price differently — i.e. if we measure price in Euros instead of Dollars? Answer: Yes, as we change the units we use on the vertical axis, slopes will change just as they do if we change units on the horizontal axis. Foundational Preliminaries: Answers to Within-Chapter-Exercises 4 Exercise 0A.19 In panel (b) of Graph 9, the supply curves of two producers who both produce x at the price p are illustrated. Which producer is more price elastic when price increases? What about when price decreases? Answer : The green producer 2 is more price elastic as price increases — raising ′′ ′ outputto x whenpriceincreasesto p whiletheblueproducer1only raisesoutput to x . Producer 2 is similarly more price elastic when price falls. Exercise 0A.20 How much does the consumer spend when price is $50? How much does she spend when price increases to $100? Answer: Whenpriceis$50, theconsumer spends50(700) =$35,000.When price is $100, the consumer spends 100(600) =$60,000. Exercise0A.21 What is the size of the blue shaded area in panel (a)? What about the magenta area? Is the difference between the magenta and the blue area the same as the increase in spending you calculated in exercise 0A.20? Answer: The blue shaded area has height of 50 and length of 100 — implying a size of 5,000. The magenta area has height of 50 and length of 600 — implying a size of 50(600)=30,000. The difference between the magenta and the blue areas is therefore 25,000. In the previous exercise we concluded that the consumer spends $35,000 at the lowerprice and $60,000 at the higher price— a difference of $25,000. Exercise 0A.22 A price ceiling is a government-enforced maximum legal price. In order for such a price ceiling to have an impact on the price at which goods are traded, would it haveto be set above or below the equilibrium price p ? ∗ ∗ Answer : If the price ceiling is set above p , the equilibrium price p is legal — and so the equilibrium is undisturbed. But if the price ceiling is set below p , ∗ ∗ the equilibrium price p is no longer legal — implying that the price ceiling has an impact. Exercise0A.23 If a priceceiling changes the priceat whichgoodsaretraded, would you expect a “shortage” or a “surplus” of goods to emerge? How would the magnitude of the shortage or surplus be related to the price elasticity of demand? Answer: A price ceiling that has an impact is set below p — where the quantity demanded read off the magenta demand curve is greater than the quantity sup- plied (read off the blue supply curve). Thus, more is demanded than supplied at the price ceiling — implying a shortage. Exercise 0A.24 A price floor is a government-enforced minimum legal price. Repeat the pre- vious two questions for a price floor instead of a price ceiling. 5 0B. Answers to Section B: Mathematical Preliminaries Answer : If the price floor is set below p , it has no impact because p is still ∗ ∗ legal. But if the price floor is set above p , the equilibrium price p is no longer legal — implying the pricefloor has animpact on the market. Ata pricefloor above p , the quantity demanded (read off the magenta demand curve) is less than the quantity supplied (readoffthe bluesupply curve) —implying thatmoreissupplied than demanded. Thus, we have a surplus. Exercise0A.25 Can you come to similar conclusions about decreases in market pricesby look- ing at Graph13? Answer: If a price decrease is accompanied by a decline in market output (as in panel (a) of Graph 13), the price decrease must be driven by a decrease in demand. But if the price decrease is accompanied by an increase in market output (as in panel (b) of Graph 13), the price decrease is driven by an increase in supply. Exercise 0A.26 Suppose that, instead of taxing the sale of x, the government subsidized con- sumer purchases of x. Thus, consumers will be paid an amount s for each good x they buy. Can you use Graph 15 to determine whether firms will benefit from such a consumer subsidy — and how the per-unit benefit for firms depends on the price elasticity of demand? Answer: Firmswillbenefitbecausetheequilibriumpricewillincrease(asshown in Graph 15 where demand shifts up because of the consumer subsidy). But firms will benefit more the more price inelastic the supply curve — because the price increase will be larger the less elastic the supply curve is. Put differently, more of the consumer subsidy will be passed onto producers as the supply curve becomes more price inelastic. Exercise 0A.27 If the goal of consumer subsidies is to raise economic output in a market, will the government be more likely to succeed in markets will high or low price elasticities of de- mand? Answer: The government will be more successful if the market supply curve is morepriceelastic (asinpanel (a)ofGraph15) thanifthe supply curveismoreprice inelastic (as in panel (b) of Graph 15). 0B Answers to Section B: Mathematical Preliminaries Exercise0B.1 Consider the function f (x,y,z) = xy +z. How would you describe this function in terms of the notation of equation (5)? What value does the function assign to the points (0,1,2), (1,2,1) and (3,2,4)? Answer: f : R →R .1 Foundational Preliminaries: Answers to Within-Chapter-Exercises 6 Exercise0B.2 How would you write the expression for the set of points that lie above the func- tion in panel (a) of Graph 16? Which is different from expression (6): the necessary or the sufficient condition? Answer: This would be written as © ª (x,y) ∈R | y ≥ x 2 . (1) + The necessary condition (which comes before the vertical line in the expres- sion) remains the same, but the sufficient condition has changed. Exercise 0B.3 Is this set a convex set? What about the set described in expression (1) and the set defined in exercise 0B.2? Answer : Yes, this is a convex set, as is the set described in equation (6) of the chapter. The set defined in the previous exercise, however, is non-convex. Exercise 0B.4 Suppose that the quantity of the good x that is demanded is a function of not only pxand I but also p ,ythe price of some other good y. How would you express such a demand function in the notation of equation (5)? 3 1 Answer: f : R →R . Exercise0B.5 Followingonexercise0B.4, supposethedemandfunctiontooktheform f (p ,p ,I)= x y (I/2)+p −yp . Hxw much of x will the consumer demand if I =100, p =20 anx p =10? y Answer: The consumer would demand (100/2)+10−2(20) =50+10−40 =20. Exercise 0B.6 Using the demand function from exercise 0B.5, derive the demand curve for when income is 100 and p =y10. Answer : Substituting I =100 and p = 10 into the function, we get f (p ) =50+ y x 10−2p = x0−2p . Invexting this, we solve the equation x = 60−2p for p to getx x p x 30−0.5x. This gives us the equation for the demand curve that has p on the x vertical axis and x on the horizontal — i.e. an equation with vertical intercept of 30 and slope of −0.5. Exercise0B.7 Verify the last sentence of the previous paragraph. Answer: From the demand function f (p ,I) = (I/x) − 10p , we get g(px) = x (200/2)−10p =10x−10p when we xubstitute in I = 200. Exercise0B.8 On a graph with p and I on the lower axes and x on the vertical axis, can you
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