C H A P T E R
Foundational Preliminaries: Answers to
0A Answers for Section A: Graphical
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
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?
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
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-
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
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
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
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
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?
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-
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
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
Answer: This would be written as
(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)?
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+
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