Three individuals have $1000 and identical preferences for gum, g, and cigarette

Question

Three individuals have $1000 and identical preferences for gum, g, and cigarettes, s, as measured by the utility function U(g,s) = 10g0.9a0.1. The price of gum is $9 and the price of cigarettes is $12. What is the market surplus/shortage at a price of $12 when the supply of cigarettes is 5?

There will be a surplus of 2/3 cigarettes.

There will be a surplus of 3 cigarettes.

There will be a shortage of 3 cigarettes.

There will be a shortage of 2/3 cigarettes.

There will be a surplus of 2/3 cigarettes.

There will be a surplus of 3 cigarettes.

There will be a shortage of 3 cigarettes.

There will be a shortage of 2/3 cigarettes.

Explanation / Answer

U = 10 x g0.9 x a0.1

Each individual's Budget line: 1000 / 3 = 9g + 12a

1000 = 27g + 36a

Consumption is optimal when MUg / MUa = Pg / Pa = 9 / 12 = 3 / 4

MUg = dU / dg = 10 x 0.9 x (a / g)0.1

MUa = dU / ds = 10 x 0.1 x (g / a)0.9

So, MUg / MUa = 9 x (a / g)

Equating with price ratio:

9 x (a / g) = 3 / 4

3a = 4g

So, 36a = 48g

Substituting in budget line:

1000 = 27g + 36a = 27g + 48g = 75g

g = 1000 / 75 = 13.33

a = 4g / 3 = 4 x 13.33 / 3 = 17.78

When Pa = 12 and a = 5,

There is a shortage of (17.78 - 5) = 12.78 cigarettes.

Please cross-check and validate your data.

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