In: Economics
Because people dislike commuting to work, homes closest to employment centers tend to be more expensive. The price of a home in a given employment center is $60 per day. The daily rental price for housing drops by $2.50 per mile for each mile farther from the employment center. The price of gasoline per mile of commute is pg (pg < $2.50) Thus, the net cost of traveling an extra mile to work is pg − $2.5. Lan chooses the distance she lives from the job center, D (where D ≤ 50), and all other goods, A. The price of A is $1 per unit. Lan’s utility function is U = (50 − D) 0.5A0.5 , and her income is Y, which for technical reasons is between 60 and 110 (60 ≤ Y ≤ 110). (a) Is D an economic bad (the opposite of a good)? To answer this question, find ∂U ∂D . (b) Draw Lan’s budget constrain (c) Use the Lagrange method to derive Lan’s demands for A and D
A)
Lan's Utility function U = (50 - D) 0.5A
First order condition for anything to maximum or minimum is
This gives us the value of A = 0
With second order condition giving us 0, the point becomes inflection point.
Distance is an economic bad as the more the distance of housing from the work place, the higher will be the cost of travelling. Even the price of house decreases as the distance of house increases from the work place. Thus, more the distance, higher the costs to Lan or other workers making distance an economic bad.
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B)
Lan's Budget Constraint will be given as
Y = PD. D + PA. A
PD = $2.5, PA = $1
and 60 < Y < 110
Y = 2.5 D + A
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C) U = (50 - D)0.5 A = 25 A - 0.5 D.A
Lagrangian Function
First order conditions
-------------------------------- eq1
--------------------------------- eq2
------------------------------- eq3
From these equations, we get the below two equations:
From eq 1 and eq2 , A = -125 +2.5 D
Substituting this in eq3 we get,
D = (Y + 125)/5
A = -62.5 + 0.5Y
where value of Y is given as 60 < Y < 110.
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