(a) Joe's optimal consumption bundle is determined by maximizing his utility subject to his budget constraint. The budget constraint is I = px*x + py*y.

To find the optimal bundle, we need to solve the following maximization problem:

Maximize U(x, y) = x^α * y^β subject to I = px*x + py*y.

This is a standard utility maximization problem that can be solved using the Lagrange method. The Lagrangian is:

L = x^α * y^β + λ(I - px*x - py*y)

Taking the derivative of L with respect to x, y and λ and setting them equal to zero gives the following first order conditions:

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

Solving these equations simultaneously gives the optimal consumption bundle (x*, y*):

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

(b) Substituting the given values into the expressions for x* and y* gives:

x* = 6/(6+2) * (24/2) = 18/2 = 9
y* = 2/(6+2) * (24/3) = 8/3 = 2.67

So, Joe's optimal choice is to consume 9 units of good x and approximately 2.67 units of good y.

(c) If px increases by 50%, the new price is px = 2*1.5 = 3. Substituting this into the expressions for x* and y* gives:

x* = 6/(6+2) * (24/3) = 12/3 = 4
y* = 2/(6+2) * (24/3) = 8/3 = 2.67

So, Joe's new optimal consumption bundle is 4 units of good x and approximately 2.67 units of good y.

The substitution effect is the change in consumption due to the change in relative prices, holding utility constant. In this case, it is the decrease in consumption of good x from 9 units to 4 units.

The income effect is the change in consumption due to the change in purchasing power caused by the price change. In this case, it is the decrease in consumption of good x from 9 units to 4 units, minus the increase in consumption of good y from 2.67 units to 2.67 units (which is zero in this case). So, the income effect is also the decrease in consumption of good x from 9 units to 4 units.

Question

(a) Joe's optimal consumption bundle is determined by maximizing his utility subject to his budget constraint. The budget constraint is I = px*x + py*y.

To find the optimal bundle, we need to solve the following maximization problem:

Maximize U(x, y) = x^α * y^β subject to I = px*x + py*y.

This is a standard utility maximization problem that can be solved using the Lagrange method. The Lagrangian is:

L = x^α * y^β + λ(I - px*x - py*y)

Taking the derivative of L with respect to x, y and λ and setting them equal to zero gives the following first order conditions:

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

Solving these equations simultaneously gives the optimal consumption bundle (x*, y*):

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

(b) Substituting the given values into the expressions for x* and y* gives:

x* = 6/(6+2) * (24/2) = 18/2 = 9
y* = 2/(6+2) * (24/3) = 8/3 = 2.67

So, Joe's optimal choice is to consume 9 units of good x and approximately 2.67 units of good y.

(c) If px increases by 50%, the new price is px = 2*1.5 = 3. Substituting this into the expressions for x* and y* gives:

x* = 6/(6+2) * (24/3) = 12/3 = 4
y* = 2/(6+2) * (24/3) = 8/3 = 2.67

So, Joe's new optimal consumption bundle is 4 units of good x and approximately 2.67 units of good y.

The substitution effect is the change in consumption due to the change in relative prices, holding utility constant. In this case, it is the decrease in consumption of good x from 9 units to 4 units.

The income effect is the change in consumption due to the change in purchasing power caused by the price change. In this case, it is the decrease in consumption of good x from 9 units to 4 units, minus the increase in consumption of good y from 2.67 units to 2.67 units (which is zero in this case). So, the income effect is also the decrease in consumption of good x from 9 units to 4 units.

Knowee AI · Accepted Answer