(a) Joe's optimal consumption bundle is found by maximizing his utility subject to his budget constraint. The budget constraint is I = px*x + py*y. We can use the method of Lagrange multipliers to solve this problem. The Lagrangian is:

L = x^α * y^β + λ(I - px*x - py*y)

Taking the partial derivatives and setting them equal to zero gives us the following first order conditions:

α * x^(α-1) * y^β - λ*px = 0
β * x^α * y^(β-1) - λ*py = 0

Dividing the first equation by the second, we get:

(α/β) * (y/x) = (px/py)

Solving for y, we get:

y = (β/α) * (x/px) * py

Substituting this into the budget constraint gives us:

I = px*x + py*(β/α) * (x/px) * py
I = px*x + β*x
x = I / (px + β)

Substituting x back into the equation for y gives us:

y = (β/α) * (I / (px + β)) * py

So Joe's optimal consumption bundle is (x, y) = (I / (px + β), (β/α) * (I / (px + β)) * py).

(b) Substituting the given values into the equations for x and y, we get:

x = 24 / (2 + 2) = 6
y = (2/6) * (24 / (2 + 2)) * 3 = 6

So Joe's optimal choice is (x, y) = (6, 6).

(c) If px increases by 50%, then the new px is 3. Substituting this into the equations for x and y, we get:

x = 24 / (3 + 2) = 4.8
y = (2/6) * (24 / (3 + 2)) * 3 = 7.2

So Joe's new optimal consumption bundle is (x, y) = (4.8, 7.2).

The income effect is the change in consumption due to the change in purchasing power caused by the price change. It is given by the difference between the new and old consumption bundles, holding utility constant. In this case, it is (4.8 - 6, 7.2 - 6) = (-1.2, 1.2).

The substitution effect is the change in consumption due to the change in relative prices, holding real income constant. It is given by the difference between the new and old consumption bundles, holding income constant. In this case, it is (4.8 - 6, 7.2 - 6) = (-1.2, 1.2).

Note: The above solution assumes that the utility function is Cobb-Douglas, which has the property that the income and substitution effects are the same. For other types of utility functions, the income and substitution effects may be different.

Question

(a) Joe's optimal consumption bundle is found by maximizing his utility subject to his budget constraint. The budget constraint is I = px*x + py*y. We can use the method of Lagrange multipliers to solve this problem. The Lagrangian is:

L = x^α * y^β + λ(I - px*x - py*y)

Taking the partial derivatives and setting them equal to zero gives us the following first order conditions:

α * x^(α-1) * y^β - λ*px = 0
β * x^α * y^(β-1) - λ*py = 0

Dividing the first equation by the second, we get:

(α/β) * (y/x) = (px/py)

Solving for y, we get:

y = (β/α) * (x/px) * py

Substituting this into the budget constraint gives us:

I = px*x + py*(β/α) * (x/px) * py
I = px*x + β*x
x = I / (px + β)

Substituting x back into the equation for y gives us:

y = (β/α) * (I / (px + β)) * py

So Joe's optimal consumption bundle is (x, y) = (I / (px + β), (β/α) * (I / (px + β)) * py).

(b) Substituting the given values into the equations for x and y, we get:

x = 24 / (2 + 2) = 6
y = (2/6) * (24 / (2 + 2)) * 3 = 6

So Joe's optimal choice is (x, y) = (6, 6).

(c) If px increases by 50%, then the new px is 3. Substituting this into the equations for x and y, we get:

x = 24 / (3 + 2) = 4.8
y = (2/6) * (24 / (3 + 2)) * 3 = 7.2

So Joe's new optimal consumption bundle is (x, y) = (4.8, 7.2).

The income effect is the change in consumption due to the change in purchasing power caused by the price change. It is given by the difference between the new and old consumption bundles, holding utility constant. In this case, it is (4.8 - 6, 7.2 - 6) = (-1.2, 1.2).

The substitution effect is the change in consumption due to the change in relative prices, holding real income constant. It is given by the difference between the new and old consumption bundles, holding income constant. In this case, it is (4.8 - 6, 7.2 - 6) = (-1.2, 1.2).

Note: The above solution assumes that the utility function is Cobb-Douglas, which has the property that the income and substitution effects are the same. For other types of utility functions, the income and substitution effects may be different.

Knowee AI · Accepted Answer