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Unformatted text preview: PRACTICE PROBLEM SET #1
ANSWER KEY Chapter 1. 2. The constraint is that the individual must achieve a level of'satisfaction, S, of
at least ten, otherwise she dies. Algebraically, this constraint can be written as (FCW2 210. ‘ The objective function is a representation of what the consumer tries to achieve.
In this case, she is trying to minimize her total expenditure on F and C (i.e., she
wants to minimize PFF + PCC). The variables under the consumer’s control are
the amounts of each good that she purchases. Therefore, the endogenous
variables are F and C, and the exogenous variables are 5, Pc, and PF. The
constrained optimization problem can be written as Min PFF + PcC
wrt F, C subject to (FC)”2 210. 3. The maximization problem is Max 100W2
wrt L subject to L 56. The solution to this problem is at point A on the graph, where the vertical line
intersects the horizontal axis at L = 6. In Other words, the vertical line
represents the constraint while the curve represents the objective. Point A
corresponds to the highest level of output that can be achieved given that total
daily work time cannot exceed 6 hours.  6) Chapter 2 2. Quantity demanded is a speciﬁc quantity on the demand curve. A change in
price will cause the quantity demanded to change (i.e., a movement along the curve). Demand is the entire curve. It represents a functional relationship
' between price and quantity demanded. A change in one of the determinants of
demand (eg. preferences, income, prices of related goods, etc.) will change
demand (i.e., shift the entire curve). Quantity supplied is a specific quantity on the supply curve. A change in price
will cause the quantity supplied to change (Le, a movement along the curve).
Supply is the entire curve. It represents a functional relationship between price
and quantity supplied. A change in one of the determinants of supply (eg. costs,
number of suppliers, sellers’ expectations of future price, etc.) will change supply
(i.e., shift the entire curve). 3. Qd = —P + 20 + I/100 Qd=Qs ‘
Pe + 20 +1/100 = 2Pe 40 + 6T
50 + I/100—6T= 3 Pa P.3 = 20 + I/300 — 2T <— Reduced Form Equation Qe = [20 + 1/300 2T] + 20 + I/100 e = 2T + 21/300 <— Reduced Form Equation I = 1200 ,
T = 4 . 4; 51 Qd=P+20+12  Qa = P + 32 <— Demand p = _Qd + 32 <— Inverse Demand Q5=2P—40+24 35”,. Q5 = 2P  16 <— Supply _ p = 1/2Qs + 8 <— Inverse Supply P,3 = 20 + I/300 — 2T
Pa = 20 ‘+ 1200/300 — 2(4) = $16 Qe = 2T + 21/300 . _
Qe = 2(4) + 2(1200)/300 = 16 If T increases to 5, this will increase supply. An increase in supply causes a
decrease in price and an increase in quantity. as QS=2P'40+30 32 Q; = 2P — 10 <— New Supply P = VZQs + 5 <— New Inverse Supply
Pe=20+I/300—2T §:=:—_ w... Pa = 20 + 12001300  2(5) = $14 Qe = 2T + 21/300
Qe = 2(5) + 2(1200)/300 = 18 If I decreases to 900 instead, this will decrease demand (since the good is a
normal good). A decrease in demand causes a decrease in price and a decrease in quantity (from the original). ‘ d:
Qd = P + 20 + 9 r 32
Qd = p + 29 <— New Demand 29
p = —Qd + 29 <— New Inverse Demand 5
Pe=lb
Pe=20+I/300—2T . /"
Pa = 20 + 900/300 — 2(4) = $15 “5 g. I
_ I
Qe = 2T + 21/300, I . D' D
Qe = 2(4) + 2(900)/300 =14 a; as a Initially, you used equilibrium analysis. Then you used comparative statics. 4. Qd=1005P
Q5 = 50 + 5P Qd = Q5
100  5Pe = 50 + SR;
lOPe = 50 P, = 5 ($500)
Qe = 100 — 5(5)
Q,. = 75 (750,000) A price ceiling at $100 (P = 1) would cause Qd (1) = 95 and Q5 (1) z ‘55.
Therefore, it creates a shortage of 40 (or 400,000) units. 5. Point price elasticity looks at elasticity at a speciﬁc price. All: price elasticity
looks at an “average" elasticity between two prices. Point price elasticity is a more accurate measure. EQdy = an/aP ' P/Qd QC] = P + 20 + 1/100 I = 1200, P = 6 EQdJ: = "1  6/26 = 0.2308 I = 1200, P = 2 EW = —1  2/30 = 0.0667 I = 1400, P = 20 EQdJ = an/BI j P/I = 1/100  (1400/14) = 1 6. de= 30—2P,+ 1.5P,—3P,+ .51 and P)( = 3, FY: 8, 19,: 4, andI= 20
de = 30 — 2(3) + 1.5(3) — 3(4) + .5(20) = 34 Ede,px = anx/an  lax/QC,x = 2  3/34 = —3/17
Ederz = anx/apz  Pz/Qdx = 3  4/34 = 45/17
Equ,1= anx/BI  I/de = 0.5  20/34 = 5/17 Ede.Pv = anx/apy  Py/de = 1.5  3/34 = 6/17 7. Because of the Law of Demand, price and quantity change in opposite
directions. Therefore, since total revenue (TR) is P  Qd, the effect of a price
change on TR depends on the relative size of the changes. The price elasticity
gives us information on the relative size of the changes. (Note that the amount
of revenue collected is the same as the amount spent. 50 consumer expenditures equal total revenue.) When demand is elastic, the quantity effect is larger than the price effect, in
percentage terms. Therefore, if price is increased, quantity demanded will
decrease by a larger percentage. So TR will fall. 0n the other, if price is
decreased, quantity demanded will increase by a larger percentage. So TR will increase. When demand is inelastic, the opposite is true. The price effect is larger than
the quantity effect, in percentage terms. Therefore, if price is increased,
quantity demanded will decrease by a smaller percentage. So TR will increase. On the other, if price is decreased, quantity demanded will increase
by a smaller percentage. 50 TR will fall. If the government is trying to raise funds through an excise tax, they will raise
more money on items whose quantity demanded does not decrease substantially
when the tax is imposed. An excise tax will increase price, and if the good has
an inelastic demand, this price increase will generate a smaller percentage
decrease in quantity demanded than if it were elastic, therefore generating more revenue. Chapter 3 2. a. Violates completeness
b. Violates nonsatiation
c. No violations
d. Violates nonsatiation 3. See your class notes or the text for the deﬁnitions. A diminishing MRS implies
that the indifference curves are convex which basically means that the slope of
the indifference curve gets ﬂatter as the individual consumes more and more of the good on the x—axis. (1) U = xx2
Mm=ﬁ MRsx, = y2/2xy = y/Zx
Convex since it decreases as x increases (and y decreases) (2) U = X1/2y1/2
MUx = (1/2)x'1/2y1/2
Muy = (1/2)x1/2y'1/2 MRSx,y = [(1/2)X'1’2y1’2]/[ (1/2)x1/2y1/2] = W
Convex since it decreases as x increases (and y decreases) (3) U=5x+3y
MUX=5
MUY=3 MRS,“ = 5/3
Not convex. The indifference curves are linear since the slope is constant. (4) U = x1’4y3’4
MUx = (1/4)x‘3/4y3/4
MUV = (3/4)x1/4y'1’4 MRSx,y = [(1/4)x3/4y3/4]/[ (3/4)X1/4y~1/4] = Y/3X
Convex since it decreases as x increases (and y decreases) (5) u = taxi/2 + 2y
MUX = 3x11”
MUy = 2 MRS” = 3x'1’2/2 = 3/2x1/2
Convex since it decreases as x increases (and y decreases) 4. If we put horror movies on the x—axis and all other goods on the yaxis, this
statement implies that the individual would be willing to give up more units of y
in order to consume additional horror movies. This would mean an increasing
MRS“. The indifference curves would therefore be concave rather than convex. All OWE?
6:22:16 H’ error
H0 v‘ I ‘58 5. See your class notes for an explanation of this. NA b. With alcohol on the horizontal axis and nonalcohol on the vertical, these
preferences would imply that for the same level of alcohol consumption, Smith’s
MRSAM would be lower than Jones’. Since Smith prefers nonalcohol to alcohol,
the number of units of nonalcohol he would be willing to give up to get alcohol (which is what MRSAM measures) is lower than what Jones, who prefers alcohol,
would be willing to give up. ...
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 Spring '08
 SCOTT

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