John has a weekly endowment of 140 dollars that he spends on buying games (G) an

Question

John has a weekly endowment of 140 dollars that he spends on buying games (G) and music cds (M). The price of each game is 30 dollars, the price of each cd is 20 dollars. He can buy any amount of the two goods that satisfies the budget constraint, including fractions (for example G=0.3987 and M=0.8765 is allowed as long as it is feasible). John’s utility is given by: U(G,M)=G1/2+M1/2

a) Find John’s Budget constraint.

b) Set up the Lagrangian and find his optimal consumption.

c) What is his new consumption if the price of games becomes 20 dollars?

Explanation / Answer

This seems easy..

a) Budget constraint will be
30G + 20M= 140
[This is because the total expenditure on games and music should not exceed $140]


b) L= G + M +( 120- 30G- 20M)

L/G = 1/2G - 30 = 0 ----- equation 1

L/M = 1/2M - 20 = 0 ----- equation 2

L/ = 120- 30G- 20M = 0

Solving equations 1 and 2, we get

M/G = 3/2

So M/G= 9/4

So, 4M= 9G

Now putting this in the budget constraint, we get

30G + 45G = 140

G = 140/75

G= 1.867

M= 4.201

c) Now price of Game is also $20 and that of Music is also $20

So we will have M/G= 1

So M/G= 1

So M= G

Putting this in budget constraint we get

20G + 20M= 140

G= 140/40

G= 3.5

M= 3.5

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