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Practice Problems Fall 2015 Econ 263
1 Budget constraints
- Each month, Andy gets the following (exogenous) endowments:
CA = 5; T (^) A = 3:
Andy can exchange these endowments for currency at market prices, which are exogenous to him, in a central marketplace.
(a) If the price of co§ee is 7 units of currency per unit of co§ee, and the price of tea is 5 units of currency per unit of tea, show why Andyís income/month measured in currency is $50.. A: CA z}|{ 5
PC z}|{ 7 +
T (^) A z}|{ 3
PT z}|{ 5 = 50:
(b) What is his income/month measured in units of co§ee? In units of tea? A: 50 7
= 7: 142 9; (C/month) 50 5
= 10 : (T/month)
(c) What is the relative price of co§ee? What are the units of this price? A: PC PT
The units are units of tea/unit of co§ee. (d) Write his budget constraint in standard slope-intercept form with consumption of tea/month on the left-hand-side of the equality sign. A: TA = 10 1 : 4 CA:
(e) With tea on the vertical axis and co§ee on the horizontal, draw a schematic diagram of his budget constraint, making sure you identify all relevant features, i.e., slope, intercepts, and endowment point. A:y = 10 1 : 4 x: Co§ee-intercept is 7 : 142 9., endowment point is (5; 3)
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
C(A)
T(A)
(f) What would happen to this schematic diagram if both PC and PT were to double? Triple? Be cut in half?
A: nothing. Both slope and intercept remain unchanged.
- Now consider another scenario. Andy grows co§ee for a living, and takes his harvest to market once a year. There, he can sell as much of his crop as he wants at a market price of a certain amount of dollars per pound of co§ee. While at the market, Andy can use the money he gets from selling his co§ee to purchase the only other good he likes to consume, tea, at a market price of a certain amount of dollars per pound of tea.
(a) Suppose Andy grows 80 pounds of co§ee per year, and co§ee ex- changes in the market place for $2.00/kilo. Tea exchanges in the market place for $4.00/kilo. Let CA symbolize the variable that measures the amount of co§ee Andy consumes per year, and TA sym- bolize the variable that measures the amount of tea Andy consumes per year. Describe in an equation with only TA on the left-hand-side of the equality sign all those pairs of kilos of co§ee/yr. and kilos of tea/year that Andy could consume at these prices, assuming he spent all of his income. Answer: Let me put the budget constraint in parametric form. First I express in an equation the equality of income and expenditure:
Expenditure z }| { PC CA + PT TA =
Income z }| { PT T (^) A + PC CA:
PC and PT , assuming he spent all of his income. In this equation, put TA as the only variable on the left-hand-side of the equality sign. Answer: TA =
80 PC
PT
PC
PT
CA:
- Let CA symbolize the variable that describes Andyís production of lbs. of co§ee per year. For arbitrary values of Andyís production of lbs. of co§ee/yr. and arbitrary values of the price of co§ee and the price of tea, describe in an equation all those pairs of pounds of co§ee/yr. and pounds of tea/yr. that Andy could consume if he spent all of his income. Again, write this equation with TA as the only variable on the left-hand-side of the equality sign. Answer: TA =
PC
PT
CA
PC
PT
CA:
- Draw a schematic diagram of the above equation in the co§ee-tea plane. By co§ee-tea plane, we mean the standard picture in which tea/year is measured on the vertical axis and co§ee/year on the horizontal axis. By schematic we mean that the key qualitative features of the equation, namely the slope and the intercepts, are depicted and identiÖed, although not necessarily to scale. Answer: slope and intercepts identiÖed, and the line connecting the in- tercepts arguably a straight line.
- What would happen to this schematic diagram if both PC and PT were to double? Triple? Be cut in half? A: nothing. Both slope and intercept remain unchanged.
2 Preferences, Demand, Equilibrium in GE Model
- 15 points. Now consider an individual (Antoine) who receives an endow- ment every month of 32 units of co§ee and 12 units of tea. He trades in a perfect market in which currency prices of co§ee and tea are symbolized as PC and PT , respectively. DeÖne the relative price of of co§ee as P PCT and symbolize this as p.
- What is Antoineís demand function when his preferences are represented by the utility functions:
UA = CATA: (1) UA = CATA + (CA)^2 (TA)^2 : (2)
UA =
CATA
2(CA + TA)
Answer: First, the tangency conditions for (1) and (2): Note they are identical.
dUA = (
CATA(2)
(2^2 (CA + TA)^2
CA(2)(CA + TA)
(2(CA + TA))^2
)dTA
2 CATA
(2(CA + TA))^2
2 TA(CA + TA)
(2(CA + TA))^2
)dCA;
dUA = 0 :
dTA dCA
2 CATA + 2TA(CA + TA)
2 CATA + 2CA(CA + TA)
2 CATA + 2CATA + 2(TA)^2
2 CATA + 2CATA + 2(CA)^2
(TA)^2
(CA)^2
= p:
Now for (3):
dUA = (
CATA(2)
(2^2 (CA + TA)^2
CA(2)(CA + TA)
(2(CA + TA))^2
)dTA
2 CATA
(2(CA + TA))^2
2 TA(CA + TA)
(2(CA + TA))^2
)dCA;
dUA = 0 :
dTA dCA
2 CATA + 2TA(CA + TA)
2 CATA + 2CA(CA + TA)
2 CATA + 2CATA + 2(TA)^2
2 CATA + 2CATA + 2(CA)^2
(TA)^2
(CA)^2
= p:
Or to put it in a form useful for graphing:
(TA)^2 = p (CA)^2 ; TA = p pCA:
Next, use the tangency conditions in the budget constraints. The budget constraint is TA = T (^) A + pCA pCA
So, for both (1) and (2),
CA z}|{ pCA =
T (^) A z}|{ 1 2
CA z}|{ 3 2 p pCA;
2 pCA =
1 + 3p 2
CAd =
1 + 3p 4 p
For (3)
(p)
(^12) (CA) =
1 + 3p 2 pCA;
CA(p + p
(^12) ) =
1 + 3p 2 CAd =
1 + 3p p + p (^12)
with such preferences have L-shaped individual indi§erence curves (with respect to the origin of the (x; y) plane) which have the right-angle apex of each curve lying along rays through the origin with equations for these rays given by:
y 1 =
1 4 3 4
x 1 =
x 1 ;
y 2 =
8 11 3 11
x 2 =
x 2 :
Each POW in this camp receives an endowment from the camp commandant of (xi; yi); i = 1; 2 :
- (a) Suppose each of the above two individuals consumes his endowment. True (T) or false (F): i. _____We can unequivocally conclude that individual one (1) is better o§ than individual two (2). ii. _____We can unequivocally conclude that individual two (2) is better o§ than individual one (1). iii. _____We can unequivocally conclude that both individuals are equally well o§. iv. _____There is no way to tell which individual is better or worse o§. Answers: FFFT. No way to make interpersonal comparisons of utility. (b) Denote the relative price of x by p, i.e., p P Pxy : Write down the competitive budget constraint for individual i. Answer: p(xi xi) = yi yi; (any variant will do, even in nominal terms). (c) With the above preferences and endowments, the demand curves for each of the above individuals is given by:
x 1 = px 1 a 1 + p
y 1 a 1 + p
; a 1 =
; (xd 1 )
x 2 = px 2 a 2 + p
y 2 a 2 + p
; a 2 =
: (xd 2 )
We get these by noting from a diagram that any most-preferred pair lies on the straight-line ray through the origin with slope, as given above. Subbing this "tangency condition" into the budget constraint then yields the above demand curves. Assume each prisoner receives endowments of one unit of x and one unit of y. Verify, i.e., show that the equilibrium conditions for this model are satisÖed, that an equilibrium price for this competitive autarkic economy is given by:
bp =
Answer: For this to be an equilibrium, aggregate demand must equal aggregate supply: x 1 + x 2 = x 1 + x 2 = 2; x 1 z }| { y 1 z}|{ 1 +
bpx 1 z }| { 11 9
1 3 +^
11 9
x 2 z }| { y 2 z}|{ 1 +
px b 2 z }| { 11 9
8 3 +^
11 9
20 9 14 9
20 9 35 9
For future reference, note that xb 1 = 107 and bx 2 = 47 :
(d) Compute the equilibrium quantities of y consumed by each individ- ual. Answer: Substitute p = 119 and xb 1 = 107 = 1: 43 ::: and xb 2 = 47 = : 57 ::: into the budget constraint, solve for:
y b 1 =
y b 2 =
For your interest, the demand curves are derived as follows from the following utility functions (known as constant elasticity of substitu- tion or CES utility functions):
Ui = [( (^) ixi) ^ + f(1 (^) i)yig ] (^
1 ); 0 < (^) i < 1; 1 1: The slope of the indi§erence curves are given by:
dy dx
y x
The tangency condition that dydx = p can be rearranged as
y = x
p(^
1+^1 )
Note bene: Take the limit of this tangency condition as ! 1 yields (remember any number raised to the zero is equal to one (1)
lim y !
x:
where individual 1 has indi§erence curves with right-angles and a "tangency condition," i.e., ray through the origin that passes through the right angles of the indi§erence curves, that are be- low those of individual 2. In autarky, each personís budget con- straint goes through the endwoment point (1; 1) and has slope 119. Now imagine rotating the budget constraint through the endowment point (1; 1) so that it now has slope 1. This is depicted in the above Ögure as the áatter budget line. Clearly, individual one is better o§ and individual two is worse o§. So, TFTF. (f) Now imagine that the camp commandant intercepts the red cross packages and reallocates the two goods so that the endowments for the two POWís are given by:
x 1 =
; y 1 =
x 2 =
; y 2 =
The two British POWís can now trade in the larger camp at the exogenous price bF T = 1: True (T) or false (F): i. _____Individual one (1) is now better o§ than in autarky. ii. _____Individual two (2) is now better o§ than in autarky. iii. _____Neither individual is now worse o§ than in autarky. iv. _____Both individuals are better o§ than in autarky. Answer: In the above diagram, imagine rotating each individ- ualís budget constraint through the autarkic consumption points. clearly, their most-preferred pairs donít change. Hence, FFTF.
4 Preferences
For each question, be prepared to explain the steps used in how you got your answer, and be prepared to illustrate these steps with appropriate schematic diagrams.
- Consider three individuals, Alex, Bobby, and Don, who have preferences represented by the following utility functions:
UA = CATA (Alexís)
UB = CB TB + (CB )^2 (TB )^2 : (Bobbyís)
UD = (CD )
(^12) (Td)
(^12) : (Donís) 5 points each. True or false with short explanation:
(a) _____If both Alex and Bobby consume two (2) units of co§ee and two (2) units of tea per unit of time, then we can say that Bobby has a higher level of satisfaction than does Alex, and Alex a higher level than Don. Answer: False. First, though, consider what happens if we simply substitute the values of co§ee and tea in each utility function:
UA = CATA =
CA z}|{ 2
TA z}|{ 2 = 4; UB = 2 2 + 2^2 22 = 20; UD =
p 2 2 = 2:
One might be tempted to say this means Bob is better o§ than Andy, and Andy is better o§ than Don, when they each consume two (2) units of co§ee and two(2) units of tea. But utility numbers are ordinal rankings, and we canít compare across individuals.
(b) _____If Alexís, Bobbyís, and Donís consumptions of co§ee and tea per unit of time increase from two (2) units of co§ee and two (2) units of tea to three (3) units of co§ee and three (3) units of tea, then we can say that Bobbyís increase in well-being is greater than Alexís increase in well-being and Alexís is greatere than Donís. Answer: false. Again, Örst consider a naive approach of considering what happens to changes in utility levels:
UA =
CA z}|{ 3
TA z}|{ 3 = 9; 9 5 = 5 utils; UB = 3 3 + 3^2 32 = 90; 90 20 = 70 utils; UD =
p 3 3 = 3; 3 2 = 1 util:
But these are not useful for making comparisons: utility numbers provide an ordinal ranking. (c) _____Bobby would be classiÖed as a tea-lover relative to Alex. Answer: false, they have the same mrs function, i.e., same slopes of indi§erence curves at every point in the co§ee-tea plane. Calcu- late the mrs^0 s: (for Bob, you need to use the "take total derivative of U and set equal to zero" approach): once without using total
unot of co§ee in exchange for tea, would require three (3) units of tea to keep him on the same indi§erence curve. That is, Bob loves co§ee so much in comparison to Andy that he needs to be compensated for the loss of one unit of co§ee with three units of tea. Bob is the co§ee-lover, relatively speaking. These indi§erence curves are plotted below, with Andyís "steeper" at (1:1):
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.
0
1
2
3
4
5
6
7
8
C
T
5 Income distribution as source of trade
- Andy and Bob are two POWís in an English camp. They have preferences speciÖed as:
UA = (CA)
(^34) (TA)
(^94) ; UB = (CB )
(^12) (TB )
(^12) :
which have known MRS functions:
M RSA =
TA
CA
; M RSB =
TB
CB
Nota bene:
UA = (CA)
(^34) (TA)
(^94) ; 3 4
1 =^
1 +^2 =^ 1;^1 + 3^1 = 1;
1 =^
UA =
(CA)
(^14) (TA)
34 ^3
UB = (CB )
(^12) (TB )
(^12) : (a) 10 points. Andy receives an endowment of one (1) unit of tea, and zero (0) units of co§ee. Bob receives an endowment of zero (0) units of tea and one (1) unit of co§ee: T (^) A = 1; CA = 0; T (^) B = 0; CB = 1: Construct the general equilibrium (GE) demand curves for co§ee for each of these individuals, and construct the market general equilib- rium demand curve. Answer: First, budget constraints:
PT
PT
TA +
PC
PT
CA =
PT
PT
1 z}|{ T (^) A +
PC
PT
0 z}|{ CA ;
PT
PT
TB +
PC
PT
CB =
PT
PT
0 z}|{ T (^) B +
PC
PT
1 z}|{ CB
Now, tangency conditions: dTA dCA
TA
CA
= p )
TA = 3pCA dTB dCB
TB
CB
= p )
TB = pCB Combine each individualís tangency condition with his or her budget constraint to get his or her demand curve:
CAd =
4 p
RA;
CBd =
2 p
RB
endowment of zero (0) units of tea, and one (1) unit of co§ee, while Baptiste receives an endowment of one (1) unit of tea and zero (0) units of co§ee:
T (^) A = 0; CA = 1; T (^) B = 1; CB = 0:
That is, the endowments have been switched relative to Andy and Bob. Construct the equilibrium demand curves for co§ee for Alphonse and Baptiste and construct the market equilibrium demand curve.
Answer:the tangency conditions remain the same:
TA = 3 pCA; TB = pCB :
The budget constraints are now:
TA + pCA = p; TB + pCB = 1 :
Upon substitution of the tangency conditions into the budget con- straints, we have
CAd =
CdB =
2 p
Aggregate or market demand:
Cd^ =
2 p
The two aggregate demand curves (one for the English and one for the French) di§er, even though aggregate endowments are the same and preferences are the same in both scenarios. We depict the two demand curves below, where the thick line represents Andy and Bobís market demand curve:
y = :25 + : 5 x ^1
0 1 2 3 4 5
0
1
2
3
4
5
C
p
The point of this exercise: aggregate (market) demand functions de- pend on income distribution. This is a caveat for most of the models we use, in which we focus on model speciÖcations that have demand functions invariant to income distribution. We focus on these models that donít have "income e§ects" because: (1) observation suggests it is not such a strong e§ect that it overturns qualitative predictions of our models; (2) we want to focus on other features without muddying the water with this complication, i.e., it has a pedagogical purpose.
(d) 5 points. Calculate the autarkic equilibrium relative price of co§ee in the French camp. Answer: Equating demand to supply: 1 4
2 p^
This implies p a =
Now, substituting this value of p back into the individual demand functions yields
C^ b A = 1 4
; Cb B =
T^ b (^) A = 1 2
; Tb (^) B =
(e) 5 points. Which of these individuals would be described as relatively co§ee-loving?
- 10 points. Assume Andyís endowment is:
CA = 3; T (^) A = 25 ;
i.e., the ordered pair (3; 25); and his preferences are represented by
UA = TA + 10CA (CA)^2 f or CA 5;
UA = TA + 25 f or CA 5 : Restrict attention to the range of values for prices such that p > 0. What is his demand curve for co§ee? For tea? Answer: For co§ee, Önd dT dCAA , set it to p; and solve:
CAd = 5
p
For tea, we need to substitute the demand for co§ee into the budget constraint:
TA = RA p
p + 5
RA z }| { 3 p + 25 +
5 p = 2 p + 25: 5
- Andyís preferences for co§ee and tea are such that two of the members of his family of indi§erence curves, named IA 800 and IA 450 , respectively, are described as:
IA 800 : TA =
CA
IA 450 : TA =
CA
(a) What are the functions that describe the slopes of each of these curves?
IA 800 :
dTA dCA
(CA)^2
IA 450 :
dTA dCA
(CA)^2
(b) Suppose CA = 80; PC = $2: 00 =lb; PT = $4: 00 =lb. Upon which of the above indi§erence curves would lie Andyís most-preferred feasible
co§ee-tea pair? By feasible we mean that he could a§ord it. Explain your reasoning. Answer: IA 800 : The most-preferred pair should be at that point on the BC at which the slope of an indi§erence curve is just equal to the slope of the BC, which is : 5. Thus,
IA 800 :
dTA dCA
(CA)^2
(CA)^2 = 1600;
CA = 40
If this is true, then the associated value of TA on the I-curve is
TA =
This pair (40; 20) is feasible (check that it satisÖes the BC). For the other I-curve,
IA 450 :
dTA dCA
(CA)^2
(CA)^2 = 900;
CA = 30 :
Hence, the associated value of TA on this I-curve is
TA =
The pair (30; 15) is less-preferred than (40; 20) because at that point there is less of both goods compared to the other feasible point.
- Suppose someone told you that two memberís of Andyís family of indif- ference curves were described by the following two equations:
IA 800 : TA =
CA
IA 20 : TA = 20 + ln
CA
If Andyís preferences satisfy the properties economistís assume in their model of the consumer, why would this person be wrong? Hint: look at the pair (40; 20). Answer: The point (40; 20) lies on both I-curves. I-curves cannot cross, i.e., share a point.
