Question

Expenditure function for perfect complements

Derive the expenditure function associated with the utility function ?(?, ?) = ???{3?, ?}

Answer 1

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Answer 2

For perfect complements, the consumer is willing to pay only for the quantity of the two goods in fixed proportions, so the utility function can be written as:

U(x, y) = min{ax, by}

where a and b are positive constants.

The expenditure function e(p_x, p_y, U) represents the minimum amount of income required to achieve a certain level of utility, U, given the prices of the two goods, p_x and p_y. We can find the expenditure function by solving the following optimization problem:

minimize p_x x + p_y y

subject to min{ax, by} = U

The Lagrangian for this problem is:

L(x, y, λ) = p_x x + p_y y + λ(U - min{ax, by})

The first-order conditions are:

∂L/∂x = p_x - λa = 0 ∂L/∂y = p_y - λb = 0 ∂L/∂λ = U - min{ax, by} = 0

From the first two conditions, we can solve for λ:

λ = p_x / a = p_y / b

Substituting into the third condition, we get:

U = min{p_x x / a, p_y y / b}

There are two cases to consider, depending on whether p_x / a is less than or equal to p_y / b:

Case 1: p_x / a ≤ p_y / b

In this case, the consumer will choose x = U/a and y = U/b, so that:

p_x x + p_y y = p_x U/a + p_y U/b = U(p_x/a + p_y/b)

Therefore, the expenditure function is:

e(p_x, p_y, U) = U(p_x/a + p_y/b)

Case 2: p_x / a > p_y / b

In this case, the consumer will choose x = U/a and y = 0, so that:

p_x x + p_y y = p_x U/a

Therefore, the expenditure function is:

e(p_x, p_y, U) = p_x U/a

To summarize, the expenditure function for the utility function ?(?, ?) = ???{3?, ?} is:

e(p_x, p_y, U) = { U(p_x/3 + p_y/?) if p_x/3 ≤ p_y/?, p_x U/3 if p_x/3 > p_y/? }