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Unformatted text preview: < α < 1 and x1 , x2 ≥ 0 are consumption of good 1 and 2 (such a
function is called CobbDouglas ). Let p1 , p2 > 0 the price of each good and
w > 0 be the wealth.
Then the consumer’s utility maximization problem (UMP) is
maximize
subject to α log x1 + (1 − α) log x2
x1 ≥ 0, x2 ≥ 0, p1 x1 + p2 x2 ≤ w. The budget set
{(x1 , x2 )  x1 ≥ 0, x2 ≥ 0, p1 x1 + p2 x2 ≤ w}
is nonempty and compact, and u(x) = α log x1 + (1 − α) log x2 is upper semicontinuous by deﬁning log 0 = −∞. Hence UMP has a solution. Since the
constraints are linear, the Guignard constraint qualiﬁcation holds. Therefore
we can apply the KarushKuhnTucker theorem. Let
L(x, λ, µ) = α log x1 + (1 − α) log x2 + λ(w − p1 x1 − p2 x2 ) + µ1 x1 + µ2 x2 ,
where λ ≥ 0 is the Lagrange multiplier corresponding to the budget constraint
p1 x1 + p2 x2 ≤ w ⇐⇒ w − p1 x1 − p2 x2 ≥ 0
and µn is the Lagrange multiplier corresponding to xn ≥ 0 for n = 1, 2. By the
ﬁrstorder condition, we get
∂
:
∂ x1
∂
:
∂ x2 α
− λp1 + µ1 = 0,
x1
1−α
− λp2 + µ2 = 0.
x2 By the complementary slackness condition, we have λ(w − p1 x1 − p2 x2 ) = 0,
µ1 x1 = 0, µ2 x2 = 0. 2014W Econ 172B Operations Research (B) Alexis Akira Toda Since log 0 = −∞, x1 = 0 or x2 = 0 cannot be an optimal solution. Hence
x1 , x2 > 0, so by complementary slackness we get µ1 = µ2 = 0. Then by the
α
ﬁrst order condition we get x1 = λα1 , x2 = 1−2 , so λ > 0. Substituting these
p
λp
into the budget constraint p1 x1 + p2 x2 = w, we get
α 1−α
1
+
= w ⇐⇒ λ = ,
λ
λ
w
so the solution is
(x1 , x2 ) = αw (1 − α)w
,
p1
p2 . Remark.
• Note that the logic goes as follows. First, we show by some
means (e.g., BolzanoWeierstrass) that an optimal solution exists. Second,
we verify that the KarushKuhnTucker theorem applies and derive the
necessary conditions for optimality. Third, since the necessary conditions
led to a unique candidate, it must be the solution.
• Since log 0 = −∞, we know that the optimal solution cannot be x1 = 0 or
x2 = 0, so the constraints x1 ≥ 0, x2 ≥ 0 never binds at the solu...
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 Winter '08
 Foster,C

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