**Unformatted text preview: **Math for Econ I- Practice Problems
New York University 1. Supply and demand
1. In a California town, the monthly charge for waste collection is $8 for 32 gallons of waste and $12.32 for 68 gallons
of waste.
(a) Find a linear formula for the cost, C, of waste collection as a function of the number of gallons of waste, w.
(b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost
of waste collection.
(c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms
of the cost of waste collection.
Solution:
(a) We ﬁnd the slope m and intercept b in the linear equation C = b + mw. To ﬁnd the slope m, we use
m= ∆C
12.32 − 8
=
= 0.12 dollars per gallon.
∆w
68 − 32 We substitute to ﬁnd b:
C = b + mw 8 = b + (0.12)(32) b = 4.16 dollars. The linear formula is C = 4.16 + 0.12w.
(b) The slope is 0.12 dollars per gallon. Each additional gallon of waste collected costs 12 cents.
(c) The intercept is $4.16. The ﬂat monthly fee to subscribe to the waste collection service is $4.16. This is the
amount charged even if there is no waste.
2. Production costs for manufacturing running shoes consist of a ﬁxed overhead of $650,000 (“ﬁxed cost”) plus a
cost of $20 per pair of shoes (“variable cost”). Each pair of shoes sells for $70.
(a) Find the total cost, C(q), the total revenue, R(q), and the total proﬁt, π(q), as a function of the number of
pairs of shoes produced, q.
(b) How many pairs of shoes must be produced and sold for the company to make a proﬁt?
Solution:
(a) The cost function is of the form C(q) = b + m · q where m is the variable cost and b is the ﬁxed cost. Since the variable cost is $20 and the ﬁxed cost is
$650,000, we get
C(q) = 650,000 + 20q.
The revenue function is of the form R(q) = pq where p is the price that the company is charging the buyer for one pair. In our case the company charges
$70 a pair so we get
R(q) = 70q.
The proﬁt function is the diﬀerence between revenue and cost, so
π(q) = R(q) − C(q) = 70q − (650,000 + 20q) = 70q − 650,000 − 20q = 50q − 650,000. (b) We are asked for the number of pairs of shoes that need to be produced and sold so that the proﬁt is larger
than zero. That is, we are trying to ﬁnd q such that π(q) > 0. Solving we get
π(q) > 0,
50q 50q − 650,000 > 0 > 650,000 q > 13,000.
!
!
Thus, if the company produces and sells more than 13,000 pairs of shoes, it will make a proﬁt.
!
3. Linear supply and demand curves are shown in Figure with price on the vertical axis.
! !
24 (a) Label the equilibrium price p0 and the equilibrium quantity q0 on the axes.
(b) Explain the eﬀect on equilibrium price and quantity if the slope, ∆p/∆q, of the supply curve increases.
ANSWER:
Illustrate your answer graphically.
(a) See Figure ??.
(c) Explain the of the on equilibrium price and quantity if the slope, ∆p/∆q, of the demand curve becomes more
(b) If the slope eﬀect supply curve increases then the supply curve will intersect the demand
negative. Illustrate your answer graphically.
curve sooner, resulting in a higher equilibrium price p1 and lower equilibrium quantity q1 .
Intuitively, this makes sense since if the slope of the supply curve increases. The amount
Solution:
produced at a given price decreases. See Figure ??.
(a) See Figure below:
p p New supply Supply p1
p0 p0
Demand q q0 1-4aq33ansafig Old supply Demand q1 q0 1-4aq33ansbfig q !
upply curve increases then the the slope Figure 1.15
Figure 1.16
(b) If supply curve the supply curve increases then the supply curve will intersect the demand curve sooner,
of will intersect the demand
ng in a higher equilibrium price p1 and lower equilibrium quantity q1 . p1 and lower equilibrium quantity q1 . Intuitively, this makes sense
resulting in a higher equilibrium price
es sense since if the slope of the if the slope ofincreases. The amount
(c) since supplyslope of the demand curve increases. Thenegative, produced atfunction will decreases. See Figure
When the curve the supply curve becomes more amount the demand a given price
price decreases. See Figure below: more rapidly and will intersect the supply curve at a lower value of q . This will
??.
decrease
1
also result in a lower value of p1 and so the equilibrium price p1 and equilibrium quantity
p
q1 will decrease. This follows our intuition, since if demand for a product lessens, the price
New supply
Supply
Old supply
and quantity purchased of the product will go down. See Figure ??. 15 p
Supply p1
p0
Demand q Demand q1 q0 1-4aq33ansbfig ! p0 q
p1
Old demand Figure 1.16
1-4aq33anscfig q1 q0 New demand the demand curve becomes more negative, the demand function will
ly and will intersect the supply curve at a lower value of q1 . ThisFigure 1.17
will
r value of p1 and so the equilibrium price p1 and equilibrium quantity
is follows our intuition, since SHORT ANSWER:
if demand for a product lessens, the price
sed of the product will go down. See Figure ??.
p
(a) q Figure 1.15 Figure 1.16 (c) When the slope of the demand curve becomes more negative, the demand function will
(c) When the slope of the demand curve becomes more negative, the demand function will decrease more rapidly
decrease more rapidly and will intersect the supply curve at a lower value of q1 . This will
and will intersect the supply curve at a lower value of q1 . This will also result in a lower value of p1 and
also result in a lower value of p1 and so the equilibrium price p1 and equilibrium quantity
so the equilibrium price p1 and equilibrium quantity q1 will decrease. This follows our intuition, since if
q1 will decrease. This follows our intuition, since if demand for a product lessens, the price
demand for a product lessens, the price and quantity purchased of the product will go down. See Figure
and quantity purchased of the product will go down. See Figure ??.
below: (a) p
Supply p0
p1
Old demand q q1 q0 New demand 1-4aq33anscfig ! Figure 1.17 Job: 1-4aq34-main Sheet: 1 Page: 1 (September 8, 2012 11 : 21) [1-4aq34-main] SHORT ANSWER:
4. A demand curvephas equation q = 100 − 5p, where p is price in dollars. A $2 tax is imposed on consumers. Find
1
the equation of the new demand curve. Sketch both curves.
Supply
1-4aq34
1. A demand curve has equation q = 100 − 5p, where p is price in dollars. A $2 tax is imposed on consumers.
Find theSolution: new demand curve. Sketch both curves.
equation of the
ANSWER:
p0
The original demand equation, q =
The original demand equation, q = 100 − 5p, tells us that 100 − 5p, tells us that Demand
Quantity demanded = 100 − 5demanded = 100 −
.
Quantity q Amount per unit
paid by consumers
q0 5 (Amount per unitpaid by consumers ) . The consumers consumers payunit + 2 dollars the price p plusbecause they pay the price p plus $2 tax. Thus, the new demand
The pay p + 2 dollars per p because they pay per unit $2 tax. Thus, the new demand
equation is
b) Equilibrium price will increase;
equation is
q = 100 − 5(p + 2) = 90 − 5p.
equilibrium quantity will decrease
q = 100 − 5(p + 2) = 90 − 5p.
See Figure 1.
(c) Equilibrium price and quantity will decrease Shifts in Supply and pDemand Curves -Taxes
20
18
Demand without tax: q = 100 − 5p Demand with tax: q = 90 − 5p 1-4aq34ansfig See Figure above. !
SHORT ANSWER:
q = 90 − 5p 90 100
Figure 1 q 5. A tax of $8 per unit is imposed on the supplier of an item. The original supply curve is q = 0.5p − 25 and the
demand curve is q = 165 − 0.5p, where p is price in dollars. Find the equilibrium price and quantity before and
after taxes.
Solution:
Before the tax is imposed, the equilibrium is found by solving the equations q = 0.5p − 25 and q = 165 − 0.5p.
Setting the values of q equal, we have
0.5p − 25
p = 165 − 0.5p
= 190 dollars Substituting into one of the equation for q, we ﬁnd q = 0.5(190) − 25 = 70 units. Thus, the pre-tax equilibrium is
p = $190, q = 70 units. The original supply equation, q = 0.5p − 25, tells us that
Quantity supplied = 0.5 (Amount per unitreceived by suppliers ) − 25.
When the tax is imposed, the suppliers receive only p − 8 dollars per unit because $8 goes to the government as
taxes. Thus, the new supply curve is
q = 0.5(p − 8) − 25 = 0.5p − 29.
The demand curve is still q = 165 − 0.5p. To ﬁnd the equilibrium, we solve the equations q = 0.5p − 29 and q = 165 − 0.5p. Setting the values of q equal,
we have
0.5q − 29 = 165 − 0.5p
q = 194 dollars
Substituting into one of the equations for q, we ﬁnd that q = 0.5(194) − 29 = 68 units. Thus, the post-tax
equilibrium is
p = $194, q = 68 units.
6. Consider the supply and demand functions
S = 3p − 50 D = 100 − 2p. (a) Sketh the graphs of the supply and demand functions.
(b) Find the equilibrium price and quantity.
(c) Suppose that a speciﬁc tax of $5 per unit is imposed upon suppliers. What are the new equilibrium price
and quantity. Justify your answer graphically. What is the eﬀect of the tax on the demander, what is it on
the supplier.
Solution:
(a) To ﬁnd the equilibrium price and quantity, we ﬁnd the point at which
Supply = Demand 3p − 50 = 100 − 2p 5p = 150 p = 30. The equilibrium price is $30. To ﬁnd the equilibrium quantity, we use either the demand curve or the supply
curve. At a price of $30, the quantity produced is 100 − 2 · 30 = 40 items. The equilibrium quantity is 40
items. In Figure below the demand and supply curves intersect at p∗ = 30 and q ∗ = 40. The equilibrium price is $30. To ﬁnd the equilibrium quantity, we use either the demand curve or the
supply curve. At a price of $30, the quantity produced is 100 − 2 · 30 = 40 items. The equilibrium
quantity is 40 items. In Figure 1.50, the demand and supply curves intersect at p∗ = 30 and q ∗ = 40.
p (price)
S(p)
p∗ = 30
D(p)
∗ fig1aq49 (b) q = 40 ! q (quantity) Figure 1.50: Equilibrium: p∗ = 30, q ∗ = 40 (c) The consumers pay p dollars per unit, but the suppliers receive only p − 5 dollars per unit because $5 goes
to the government as taxes. Since
Quantity supplied = 3(Amount per unit received by suppliers) − 50,
the new supply equation is
Quantity supplied = 3(p − 5) − 50 = 3p − 65;
the demand equation is unchanged:
Quantity demanded = 100 − 2p.
At the equilibrium price, we have
Demand = Supply 100 − 2p = 3p − 65 165 = 5p p = 33. The equilibrium price is $33. The equilibrium quantity is 34 units, since the quantity demanded is q =
100 − 2 · 33 = 34.
Notes: In this example, the equilibrium price was $30; with the imposition of a $5 tax in Example ??, the
equilibrium price is $33. Thus the equilibrium price increases by $3 as a result of the tax. Notice that this
is less than the amount of the tax. The consumer ends up paying $3 more than if the tax did not exist.
However the government receives $5 per item. The producer pays the other $2 of the tax, retaining $28 of
b: bchap1-temp Sheet: 37 Page: 37 (August 15, 2012 10 : 14) [ex-1-4] the tax was imposed on the producer, some of the tax is passed on to
the price paid per item. Although
the consumer in terms of higher prices. The tax has increased the price and reduced the number of items
sold. See Figure below. Notice that the taxes have the eﬀect of moving the supply curve up by $5 because
suppliers have to be paid $5 more to produce the same quantity.
1.4 APPLICATIONS OF FUNCTIONS TO ECONOMICS 37 p (price paid by consumers)
Supply: With tax
Supply: Without tax 33
30
28
Demand
fig1aq50 34 40 q (quantity) Figure 1.51: Speciﬁc tax shifts the supply curve, altering the equilibrium price and quantity
! Budget Constraint
An ongoing debate in the federal government concerns the allocation of money between defense
and social programs. In general, the more that is spent on defense, the less that is available for
social programs, and vice versa. Let’s simplify the example to guns and butter. Assuming a constant
budget, we show that the relationship between the number of guns and the quantity of butter is
linear. Suppose that there is $12,000 to be spent and that it is to be divided between guns, costing nstraint n ongoing debate in the federal governmentin the federalallocation of money betweenallocation of money between defense and social pro7. An ongoing debate concerns the government concerns the defense
d social programs. In grams. In general, the more on defense, theon defense, available that is available for social programs, and vice
general, the more that is spent that is spent less that is the less for
cial programs, and vice versa. Let’s simplify the example to guns andand butter. Suppose that there is 12,000 dollars to be spent and that
versa. Let’s simplify the example to guns butter. Assuming a constant
dget, we show that the relationship between guns, costing $400 and the quantity of costingis$2000 a ton. Therefore, the equation of the
it is divided between the number of guns each, and butter, butter
ear. Suppose that there is $12,000budget constraint is 400g + 2000b = between guns, costing
company’s to be spent and that it is to be divided 12, 000.
00 each, and butter, costing $2000 a ton. Suppose guns as a functionbought is g, and the number
(I) Write the number of the number of guns of the tons of butter.
tons of butter is b. Then the amount offunction from part (I) (which and the amountb, is on the horizontal axis?)
(II) Graph the money spent on guns is $400g, variable, g or spent on
tter is $2000b. Assuming all the money is spent,
Solution: (I) As stated in the problem,
Amount spent on guns + Amount spent on butter =guns + Amount spent on butter = $12,000
Amount spent on $12,000 or 400g + 2000b = 12,000. 400g + 2000b = 12,000. Thus, dividing both sides by 400,
hus, dividing both sides by 400, g + 5b = 30.
g + 5b = 30.
This equation is the budget constraint. Since the budget constraint can be written as
his equation is the budget constraint. Since the budget constraint can be written as
g = 30 − 5b,
g = 30 − 5b,
the graph of the budget constraint is a line. e graph of the budget constraint is a line. See Figure 1.52.
(II)
g (number of guns)
30
g + 5b = 30 fig1aq51 Section 1.4 6 ! b (tons of butter) Figure 1.52: Budget constraint 8. A person has m dollars to spend on the purchase of two commodities. The prices for commodities are p and q
dollars per unit. Suppose x units of the ﬁrst and y units of the second commodity are bought. Write a formula
to describe the budget of the buyer.
Answer: m = px + qy
9. If a ﬁrm sells Q tons of a product, the price P in dollars, received per ton is P = 1000 − Q. The price it has to
pay per ton is P = 800 + Q. In addition, it has transportation costs of 100 dollars per ton.
(a) Express the ﬁrm’s proﬁt π as a function of Q, the number of tons sold, and ﬁnd the proﬁt maximizing
quantity.
(b) Suppose the government imposes a tax on the ﬁrm’s product of 10 dollars per ton. Find the new expression
for the ﬁrms proﬁts π and the new proﬁt maximizing quantity.
ˆ
Answer:
(a) The proﬁt function is π(Q) = Q(1000 − Q) − (Q(800 + Q)) − 100Q = −2Q2 + 100Q. To ﬁnd the proﬁt
maximizing quantity we complete the square:
−2(Q2 − 50Q) = −2((Q − 25)2 − 252 ) = −2(Q − 25)2 + 1250.
Hence the proﬁt maximizing quantity is 25 tons and the maximum proﬁt is 1250 dollars. (b) The new proﬁt function is π(Q) = Q(1000 − Q) − (Q(800 + Q)) − 110Q = −2Q2 + 90Q. To ﬁnd the proﬁt
maximizing quantity we complete the square:
−2(Q2 − 45Q) = −2((Q − 22.5)2 − 22.52 ) = −2(Q − 22.5)2 + 1012.5.
Hence the proﬁt maximizing quantity is 22.5 tons and the maximum proﬁt is 1012.5 dollars.
10. (Textbook Section 4.6 Exercise 8-Harder Problem-Same ideas as in the previous problem, instead of numbers you
have constants. I want you feel comfortable with these.)
If a cocoa shipping ﬁrm sells Q tons of cocoa in England, the price received is given by P = α1 − 1 Q. On the
3
1
other hand, if it buys Q tons from its only source in Ghana, the price it has to pay is given by P = α2 + 6 Q. In
addition it costs γ per ton to ship cocoa from its supplier in Ghana to its customers in England (its only market).
The numbers α1 , α2 and γ are all positive.
(a) Express the cocoa shipper’s proﬁt as a function of Q, the number of tons shipped.
(b) Assuming that α1 − α2 − γ > 0, ﬁnd the proﬁt maximizing shipment of cocoa.
(c) Suppose the government of Ghana imposes an export tax on cocoa of t per ton. Find the new expression
for the shipper’s proﬁt and the new quantity shipped?
(d) Calculate the government’s export tax revenue as a function of t, and advise it on how to obtain as much
tax revenue as possible.
Solution:
(a) The proﬁt function is π(Q) = Q(α1 − 1 Q) − (Q(α2 + 1 Q)) − γQ = − 1 Q2 + (α1 − α2 − γ)Q.
3
6
2
(b) To ﬁnd the proﬁt maximizing quantity we complete the square:
1
1
1
1
− (Q2 −2(α1 −α2 −γ)Q) = − ((Q−(α1 −α2 −γ))2 −(α1 −α2 −γ)2 ) = − (Q−(α1 −α2 −γ))2 + (α1 −α2 −γ)2
2
2
2
2
1
Hence the proﬁt maximizing quantity is (α1 − α2 − γ) and maximum proﬁt is 2 (α1 − α2 − γ)2 . 1
(c) The new proﬁt function is π(Q) = Q(α1 − 3 Q) − (Q(α2 + 1 Q)) − γQ − tQ = − 1 Q2 + (α1 − α2 − γ − t)Q.
6
2
To ﬁnd the proﬁt maximizing quantity we complete the square: 1
1
− (Q2 − 2(α1 − α2 − γ − t)Q) = − ((Q − (α1 − α2 − γ − t))2 − (α1 − α2 − γ − t)2 ).
2
2
Hence the proﬁt maximizing quantity is (α1 − α2 − γ − t).
(d) The governments export tax revenue on the proﬁt maximizing quantity is
R = t ∗ Q∗ = t(α1 − α2 − γ − t) = −t2 + (α1 − α2 − γ)t.
To ﬁnd t that maximizes tax revenue we need to complete the square.
1
1
R = −(t2 − (α1 − α2 − γ)t) = − (t − (α1 − α2 − γ))2 − (α1 − α2 − γ)2
2
2
Hence revenue will be maximized if the government charges t = 1 (α1 − α2 − γ) dollars per ton.
2 11. (Bertrand Model of Price Competition) If there is more than one producer/seller in a market but not so many as
to make the perfectly competitive model applicable, we say that the market structure is oligopolistic. One model
that describes behavior of ﬁrms in this setting is the so-called Bertrand model. To make the matters simple, we
assume that there are two ﬁrms in the market. We assume that two ﬁrms compete in prices. Each ﬁrm sets
a price and then meets whatever demand exists for its product at that price. We assume that they produce
identical commodities, if one ﬁrm charges a lower price than the other, then all the consumers will purchase from
that producer. If the two ﬁrms charge the same price, then we assume that consumers’ purchases will be split
evenly between two producers. Thus we need to think about how revenue for each ﬁrm changes at prices that
are altered. Let’s suppose that the demand function is y = 20 − 2p and the cost function is C = 4y. Analyse
ﬁrm 1’s proﬁt and revenue functions for alternative prices given that a speciﬁc price has been set by ﬁrm 2. For
simplicity ﬁrst assume that ﬁrm 2 sets a price of p = 7 dollars. Then generalize this to any price.
Solution: Suppose ﬁrm 2 charges p2 = 7 dollars. If p1 > 7 then the revenue of ﬁrm 1, R1 = 0. If p1 = 7 then
they share the market. Since for p = 7 the demand is y = 20 − 2(7) = 6 units ﬁrm 1 will get only 3 of it hence
its revenue is then 3 · 7 = 21 dollars. Also since the cost of producing 3 units is C = 4 · 3 = 12, ﬁrm 2’s proﬁt
for p1 = 7 will be 21 − 12 = 9 dollars. If p1 < 7 dollars then ﬁrm 1 will get everybody, hence it’s revenue is
R1 = p1 (20 − 2p1 ). So for this special case when we assume that p2 = 7 dollars we get the following revenue, and
proﬁt functions for company 1: p1 (20 − 2p1 ) R1 (p1 ) = 21 0 if p1 < 7
if p1 = 7
if p1 > 7 p1 (20 − 2p1 ) − 4(20 − 2p1 ) π1 (p1 ) = 9 0 if p1 < 7
if p1 = 7
if p1 > 7 Now this can be easily generalized to a case where the set price for ﬁrm 2 is p2 . p1 (20 − 2p1 ) 1
R1 (p1 ) = 2 p1 (20 − 2p1 ) 0 if p1 < p2
if p1 = p2
if p1 < 7 p1 (20 − 2p1 ) − 4(20 − 2p1 )
π1 (p1 ) = 1 (p1 (20 − 2p1 ) − 4(20 − 2p1 ))
2 0 if p1 < p2
if p1 = p2
if p1 > p2 2-main Sheet: 1 Page: 1 (August 25, 2014 10 : 57) [1-4aq2-main]
Job: bchap1-temp Sheet: 38 Page: 38 (August 15, 2012 10 : 14) [ex-1-4] 2. Exponential growth, present values, completing the square
1 1. In Figure below, which shows the cost and revenue functions for a product, label each of the following:
38
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