Econ 8100 Solution
Bruce A. Seaman
Answers to Part III; Exam I; Spring 1997
a. X = 312.5; Y = 500. These are derived by utilizing the first order condition:
Ration of marginal utilities = ratio of prices, which via the steps I reiterated
in class yield (along with the budget constraint) demand functions of the standard
Cobb-Douglas form: X = M/2Px and the same form for Y. Given M = $2,500
and the prices of $4 and $2.50, the quantities at the tangency point where utility
is maximized are easily derived.
b. The Slutsky compensating variation in income, which will in this case be negative (since the price of X declined, so we must reduce income to keep real income constant) is easily obtained by inserting the new price of X in the budget constraint at the original optimal quantities to get:
$3.5 (312.5) + $2.5 (500) = $2,343.75, which is $156.25 lower than the original
M of $2,500, hence the Slutsky result is $ - $156.25.
The Hicksian (negative) compensating variation in income will in this case
be a larger negative value than the Slutsky, since with a price reduction for
X, applying only the Slutsky adjustment will leave the person better off when
that person substitutes toward the now relatively lower priced X. To derive
the Hicksian result, derive the utility along the original indifference curve
which is U = XY, or U = (312.5)(500) = 156,250.
Now, note that while Y = 1.6X given the original relative prices, Y = 1.4X
at the new relative prices (MU(x)/MU(y) = P(x)/P(y); or Y/X = $3.5/$2.5; Y =
1.4X). Substitute for Y in the utility function to get U = (X) (1.4X) = 156,250
= 1.4 X 2 ; X = 334.08; Y = 467.71.
At this new tangency point along the original indifference curve, the required
expenditure at the new prices is $3.5 (334.08) + $2.5 (467.71) = $2,338.55,
which is $161.45 less than the original M of $2,500. Hence, the Hicksian compensating
variation in income is $ - 161.45, a modestly larger adjustment than the Slutsky
result.
c. The full effect of the price change in X without any variation in income
is to increase X from 312.5 to 357.14, which is derived simply by plugging the
lower price of X back into the demand function identified in part (a), i.e.
$2,500/2($3.5) = 357.14. Of this total change in X of 44.64 units, the substitution
effect is from 312.5 to the new tangency point X derived in (b) of 334.08, or
a change of 21.58. The remainder of the change occurs as you move to the higher
indifference curve that the lower price (without any actual compensating reduction
in income) allows, at the X = 357.14, so the real income effect is 357.14 -
334.08 = 23.06. The full effect is thus accounted for, i.e. 44.64 = 21.58 +
23.06.
Note that one could have approximated these effects less accurately by using
the Slutsky compensating variation in income, which would have then required
you to derive the new X when M was equal to $2,500 - $156.25 (the Slutsky compensating
variation in income) = $2,343.75, at the new price of X = $3.5, which would
have yielded X = $2,343.75/ (7) = 334.82. Then the substitution effect would
be 334.82 - 312.5 = 22.32, and the real income effect would have been 357.14
- 334.82 = 22.32, which in this case happens to be equal to the substitution
effect. There is no general result that would make those two effects equal when
using the Slutsky approach to separating out the two effects.. It is, however,
a general result that for a price reduction in X, the Slutsky approach will
attribute more of the full effect on X to the substitution effect (here, 22.32
vs. the more accurate Hicksian 21.58), and less of the full effect to the real
income effect (here, 22.32 vs. the Hicksian 23.06) (Question: can you determine
how these relative effects would work in the case of a price increase
instead of a price reduction?)
Note that in this problem the differences between the Hicksian and the Slutsky
way of separating the effects is small, but can be somewhat larger in other
problems. Again, the Hicksian approach is theoretically the more accurate since
it really does keep utility constant, but the Slutsky approach is more feasible
(i.e. does not require knowledge of the utility function).
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