(x 1 ,x 2 ) MRS(x 1 ,x 2 )= - p1 p2 MRS - ∂u(x 1 ,x 2 )/∂x 1 ∂u(x 1 ,x 2 )/∂x 2 = - p 1 p 2 ⇒ ∂u(x 1 ,x 2 )/∂x 1 ∂u(x 1 ,x 2 )/∂x 2 = p 1 p 2 u(x 1 ,x 2 )= x 1 x 2 + x 1 x 2 +1 x 1 = p 1 p 2 (6, 2) 2+1 6 = p 1 p 2 ⇒ 1 2 = p 1 p 2 = ⇒ p 2 =2p 1 6p 1 + 2(2p 1 ) = 100 ⇒ 10p 1 = 100 ( p 1 = 10 p 2 = 20
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Consumer choice - Fonaments de l'Anàlisi Econòmicapareto.uab.es/prey/micro_3_ingl__s.pdfConsumer choice 27 de octubre de 2011 3.1 Optimal choice (x 1;x 2) must meet the condition
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Consumer choice
27 de octubre de 2011
3.1
Optimal choice (x1, x2) must meet the condition MRS(x1, x2) = −p1p2 .MRS is the ratio of the derivatives of the utility function, then we have:
−∂u(x1, x2)/∂x1∂u(x1, x2)/∂x2
= −p1p2⇒ ∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2=p1p2
If utility function is u(x1, x2) = x1x2 + x1, then :
x2 + 1
x1=p1p2
Substituting the conssumption bundle (6, 2) :
2 + 1
6=p1p2⇒ 1
2=p1p2
=⇒ p2 = 2p1
Substituting in budget constraint who must also meet the optimal choise:
6p1 + 2(2p1) = 100⇒ 10p1 = 100
{p1 = 10
p2 = 20
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3.2
1.
Budget set: 2x1 + 3x2 5 10
Budget constraint: 2x1 + 3x2 = 10
2.
The consumer's utility is u(x1, x2) = x1 + x2 , good 1 gives thesame utility that good 2. That's because both goods are perfectsubstitutes (MRS is -1). However, the price of goods 1 and 2aren´t the same: good 1 costs 2 monetary units while the priceof good 2 is 3 monetary units. That explain why the optimalchoice will be (5, 0).
Conssumption bundle (5, 3) can't be optimal because is ouside thebudget set, the consumer doesn't have enough money to accessit.
Conssumption bundle (2, 2) is not optimal because the utility ofthis conssumption bundle (u(x1, x2) = 2 + 2 = 4) is less than theoptimal bundle (5, 0) where the value is 5.
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3.
If prices where (5, 5), the slope of the budget constraint and theMRS would be the same (−1), then any point of the budgetconstraint would be an optimal solution.
3.3
1. With card system, our budget set will be: 80x1 + x2 ≤ 50000.
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2. For the second tari�, our budget set will be: 40x1 + x2 = 48000.
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3.
With the �rst option; 80x1+x2 ≤ 50000, for each unit of x1 we could buy80 units of x2. Then:
If α < 80⇒ x∗1 = 0 y x∗2 = 50000⇒ u(x∗1, x∗2) = 50000
If α = 80 , optimal conssumption will be any point of budget cons-traint ⇒ u(x∗1, x
∗2) = 50000
If α > 80⇒ x∗1 = 625 y x∗2 = 0⇒ u(x∗1, x∗2) = 625α
With the �rst option; 40x1+x2 ≤ 48000, for each unit of x1 we could buy40 units of x2. Then:
If α < 40⇒ x∗1 = 0 y x∗2 = 48000⇒ u(x∗1, x∗2) = 48000
If α = 40 , optimal conssumption will be any point of budget cons-traint ⇒ u(x∗1, x
∗2) = 48000
If α > 40⇒ x∗1 = 1200 y x∗2 = 0⇒ u(x∗1, x∗2) = 1200α
2. If his preferences were of the type u(x1, x2) = x1x2, the consumer chooseplan B since he has no limitation on the conssumption of good 1 and thereforecould get an in�nite utility.