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Answer Sheet, Midterm Exam # 1
1.a) Set up the Lagrangian: L ( x, y, λ ) = xy 2 + λ [30 − x − 2 y ] .
FOCs:
2
2
∂L
= y* − λ * = 0 ⇔ y* = λ *
∂x
∂L
= 2 x* y* − 2λ * = 0 ⇔ 2 x* y* = 2λ *
∂y
∂L
= 30 − x* − 2 y* = 0 ⇔ x* + 2 y* = 30
∂λ
2
2
y*
λ*
y*
1
Divide the first equation by the second: * * = * ⇔ * * = . This is our standard
2x y
2λ
2x y
2
tangency condition that the indifference curve and the budget constraint are equally
steep, i.e., that the MRS (on the left-hand side) must be equal to the price ratio (on the
right-hand side). The tangency condition gives us the optimal relative mix of xylophones
and yoyos, and therefore the optimal amount of yoyos as a function of the optimal
2
y*
1
y*
amount of xylophones: * * = ⇔ * = 1 ⇔ y* x* = x* . Substitute this expression
2x y
2
x
into the third FOC equation to solve for the optimal amount of xylophones:
x* + 2 y * x* = 30 ⇔ ⇔ x* + 2 x* = 30 ⇔ 3 x* = 30 ⇔ x* = 10 . Finally, substitute this into
( )
( )
the expression for the optimal amount of yoyos as a function of the optimal amount of
xylophones: y* x* = x* ⇔ y* = 10 .
( )
An alternative approach, which would save us a little bit of time, is to by-pass the
Lagrangian and instead go straight to the tangency condition and then solve from there
just as we did above.
b) If he consumes 10 xylophones and 10 yoyos, then Daniel achieves a utility of
(
)
2
U x* , y* = x* y * = 10 ⋅102 = 1000 . Since Daniel’s utility is ordinal rather than cardinal,
this number tells us nothing meaningful about how happy his optimal consumption
makes him.
c) No. Daniela’s preferences are different from Daniel’s because her utility function is
not a positive monotone transformation of his. Therefore, we would expect her to choose
a different optimal consumption bundle.
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d) No. Because she chooses a different consumption bundle and has a different utility
function, there is no reason to believe that Daniela’s utility will be the same as Daniel’s
when they both consume their optimal consumption bundles. Moreover, even if the two
did have the same utility, this would tell us nothing interesting since their utilities are
ordinal.
e) Yes. Since they both have Cobb-Douglas preferences, Daniel and Daniela will both
choose interior consumption bundles with tangency between budget constraint and an
indifference curve. Because they face the same prices, this means that at their respective
optimal consumption bundles, their indifference curves will be equally steep, i.e., their
marginal rates of substitution will be identical.
2.a) True. If one good is desirable and the other is not, then if we give the consumer a
little more of one of the goods, the off-setting change in the quantity of the other good
that prevents her utility from changing is in the same direction. If the good that increased
in quantity was the desirable one, then to make her worse off we must add some of the
undesirable good. If the good that increased in quantity was the undesirable one, then to
make her better off we must add some of the desirable good.
b) True. For a smooth curve, strict convexity is the same as a positive second derivative,
i.e., the MRS must be decreasing.
c) False. Because Cobb-Douglas preferences are smooth, a change in the price ratio will
move the point of tangency and therefore change the relative mix of the two goods in the
optimal consumption bundle.
d) True. With Cobb-Douglas preferences, it is optimal to split any budget in the same
way so that the relative expenditure on the two goods is identical for all income levels.
The relative expenditure on each good reflects their relative ability to generate happiness
for the consumer: the even split of expenditure across the two goods tells us that this
particular consumer has Cobb-Douglas preferences with α = β .
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3. Laura has perfect-complements preferences. In order not to waste any utilitygenerating ability of nuts and bolts, Laura must consume so that both arguments of her
minimum-operator bind (graphically, this will be at the straight-angle kink in her
indifference curves). Algebraically, this means that she should consume where N * = B* ,
i.e., she should consume as many nuts as bolts. Substitute this expression for the quantity
of nuts as a function of the number of bolts into the budget constraint and solve for the
optimal number of bolts. N * + 3B* = 20 ⇔ B* + 3B* = 20 ⇔ 4 B* = 20 ⇔ B* = 5 .
Substitute this back into the expression for the optimal number of nuts as a function of
the number of bolts to derive the obvious, namely that N * = B* = 5 .
b) The one-to-one ratio of nuts to bolts is still obvious. However, with the higher price,
Laura must decrease her quantity of both: 2 N * + 3B* = 20 ⇔ 2 B* + 3B* = 20 ⇔
⇔ 5 B* = 20 ⇔ B* = 4 ⇒ N * = 4 .
c) With the new prices, the old consumption bundle costs $2 ⋅ 5 + $3 ⋅ 5 = $25 , so in order
to compensate her for the price increase, her income would have to increase by $5.
4.a) Any positive monotone transformation will do. The simplest ones would be to add a
constant, k, or multiply by a strictly positive number, c:
U ' ( x, y ) = cU ( x, y ) + k = cx 2 y 2 + k .
b) Indifference curves remain unchanged if the MRS remains unchanged. Therefore, we
need to show that the MRS is the same for both utility functions. Start with the original
MU x 2 xy 2 y
one. Its MRS is equal to
=
= . The MRS of the new utility function is equal
MU y x 2 2 y x
MU x' c 2 xy 2 y
=
= . Since the two utility functions generate the same MRS they must
MU y' cx 2 2 y x
represent the same preferences.
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