consumer choice - fonaments de l'anàlisi...
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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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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
Therefore:
If α ≤ 40:
Choose 1º option: x∗1 = 0, x∗2 = 50000⇒ u(x∗1, x∗2) = 50000
If 40 < α ≤ 500001200 :
Choose 1º option: x∗1 = 0, x∗2 = 50000⇒ u(x∗1, x∗2) = 50000
If α < 500001200 :
Choose 2º option: x∗1 = 1200, x∗2 = 0⇒ u(x∗1, x∗2) = 1200α
3.4
1.Budget set of Plan A:{20x1 + 20x2 ≤ 8000, si x1 ≤ 200
(20 · 200) + 10(x1 − 200) + 20x2 ≤ 8000, si x1 > 200⇒
{20x1 + 20x2 ≤ 8000, si x1 ≤ 200
10x1 + 20x2 ≤ 6000, si x1 > 200
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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.
3.
u(x1, x2) = min
{x1,
1
2x2
}At the optimum we know that:
x1 =1
2x2 ⇒ x∗2 = 2x∗1
With the plan A:
1rst stretch: 20x1 + 20x2 = 8000
x∗1 =400
3, x∗2 =
800
3
u(400
3,800
3) = min
{400
3,800
3
}=
400
3
2ndstretch: 10x1 + 20x2 = 6000
x1 = 120 x2 = 240⇒ pero 120 < 240!︸ ︷︷ ︸contradiction!
With the plan B:
6000 + 20x2 = 8000⇒ x∗2 = 100
x2 = 2x1 ⇒ x∗1 = 50
u(50, 100) = min {50, 100} = 50
Then:
Like400
3> 50, will choose the planA!
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3.5
1.Maxx1,x2
x1/41 x
3/42
s.a. 100x1 + 2500x2 ≤ 40000
L(x1, x2, λ) = x1/41 x
3/42 − λ(100x1 + 2500x2 − 40000)
∂L∂x1
= 0⇒ 14x
−3/41 x
3/42 − 100λ = 0
∂L∂x2
= 0⇒ 34x
1/41 x
−1/42 − 2500λ = 0
∂L∂λ = 0⇒ 100x1 + 2500x2 − 40000 = 0
We divide the �rst ecuation by the second:
1/4x−3/41 x
3/42
3/4x1/41 x
−1/42
=100λ
2500λ
x23x1
=100
2500
x2x1
=300
2500
x1 =25
3x2
Substituing x1 in the budget constraint and found x2:
100(25
3x2) + 2500x2 − 40000 = 0
100(25
3x2) +
7500
3x2 −
120000
3= 0
2500x2 + 7500x2 = 120000
x2 = 12
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Optimal conssumption will be:{xopt1 = 100
xopt2 = 12
Graphical representation:
2.Maxx1,x2
x1/41 x
3/42
s.a. 100x1 + 2250x2 ≤ 39900
L(x1, x2, λ) = x1/41 x
3/42 − λ(100x1 + 2250x2 − 39900)
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∂L∂x1
= 0⇒ 14x
−3/41 x
3/42 − 100λ = 0
∂L∂x2
= 0⇒ 34x
1/41 x
−1/42 − 2250λ = 0
∂L∂λ = 0⇒ 100x1 + 2250x2 − 39900 = 0
We divide the �rst ecuation by the second:
1/4x−3/41 x
3/42
3/4x1/41 x
−1/42
=100λ
2250λ
x23x1
=100
2250
x2x1
=300
2250=
2
15
x1 =15
2x2
Substituing x1 in the budget constraint and found x2:
100(15
2x2) + 2250x2 − 39900 = 0
750x2 + 2250x2 − 39900 = 0
x2 =39900
3000
x2 = 13,3
Optimal conssumption will be:{xopt
′
1 = 99,75
xopt′
2 = 13,3
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Graphical representation:
3. We need to �nd were the optimal conssumption produces more utility:
u(xopt1 , xopt2 ) = 1001/4 · 123/4 ' 20,38
u(xopt′
1 , xopt′
2 ) = 99,751/4 · 13,33/4 ' 22
u(xopt′
1 , xopt′
2 ) > u(xopt1 , xopt2 )
Yes, the consumer will become a member.
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3.6
1. Budget constraint is: 2x1 + 5x2 = 40. In this case MRS is − 83 while the
slope of the budget constraint is − 25 . Optimal choice will be (20, 0) and is on
the budget constraint: 2(20) + 5(0) = 40.
2.MRS(x1, x2) = −
p1p2
−12x1x
−1/22 · x2
12x1x
−1/22 · x1
= −2
5
2x1 = 5x2
x1 =5
2x2
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Substituing in budget constraint:
2(5
2x2) + 5x2 = 40
x2 = 4
Optimal choice will is (10, 4) and is on budget constraint:
2(10) + 5(4) = 40
3.Optimal conditions are: {
x1 = 3x2
2x1 + 5x2 = 40{x1 = 3x2
2(3x2) + 5x2 = 40{x1 = 120/11
x2 = 40/11
Optimal choice is (120/11, 40/11) and is on the budget constraint:
2(120/11) + 5(40/11) = 40
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