(a) Collin's utility function is U(m, s) = 2√m + s. His budget constraint is 100 = 2m + 10s. To find his optimal consumption bundle, we need to maximize his utility subject to his budget constraint.

Setting up the Lagrangian, we get: L = 2√m + s + λ(100 - 2m - 10s)

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

dL/dm = 1/√m - 2λ = 0
dL/ds = 1 - 10λ = 0
dL/dλ = 100 - 2m - 10s = 0

Solving these equations simultaneously gives the optimal consumption bundle (m*, s*):

m* = 25
s* = 5

(b) Unfortunately, I can't draw a graph here, but I can describe how to do it. On the horizontal axis, you have m (milkshakes) and on the vertical axis, you have s (sushi). The budget line is a straight line with a slope of -2/10 = -1/5, and it intersects the m-axis at m = 50 and the s-axis at s = 10. The indifference curve for the utility level U(m*, s*) = 2√25 + 5 = 15 is a curve that is increasing at a decreasing rate. The optimal bundle (m*, s*) = (25, 5) is the point where the budget line is tangent to the indifference curve.

(c) If the price of milkshakes increases to $5, the new budget constraint is 100 = 5m + 10s. Solving this constraint for s gives s = 10 - 0.5m. Substituting this into the utility function gives U(m, s) = 2√m + 10 - 0.5m. Taking the derivative of this with respect to m and setting it equal to zero gives the new optimal consumption bundle (m*, s*):

m* = 16
s* = 2

(d) Again, I can't draw a graph here, but the new budget line has a slope of -5/10 = -1/2, and it intersects the m-axis at m = 20 and the s-axis at s = 10. The new optimal bundle (m*, s*) = (16, 2) is the point where the new budget line is tangent to the indifference curve for the utility level U(m*, s*) = 2√16 + 2 = 10.

(e) 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 milkshakes from 25 units to 16 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 milkshakes from 25 units to 16 units, minus the increase in consumption of sushi from 5 units to 2 units, which is a decrease of 7 units.

(f) The amount of additional income needed for Collin to achieve the initial level of utility is the amount that would allow him to purchase the initial bundle at the new prices. This is 5*25 + 10*5 = $175. The amount of additional income needed for Collin to purchase the initial bundle is the difference between this and his current income, which is $175 - $100 = $75. The ideal cost of living index is the ratio of the new cost of the initial bundle to the old cost, which is $175/$100 = 1.75.

Question

(a) Collin's utility function is U(m, s) = 2√m + s. His budget constraint is 100 = 2m + 10s. To find his optimal consumption bundle, we need to maximize his utility subject to his budget constraint.

Setting up the Lagrangian, we get: L = 2√m + s + λ(100 - 2m - 10s)

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

dL/dm = 1/√m - 2λ = 0
dL/ds = 1 - 10λ = 0
dL/dλ = 100 - 2m - 10s = 0

Solving these equations simultaneously gives the optimal consumption bundle (m*, s*):

m* = 25
s* = 5

(b) Unfortunately, I can't draw a graph here, but I can describe how to do it. On the horizontal axis, you have m (milkshakes) and on the vertical axis, you have s (sushi). The budget line is a straight line with a slope of -2/10 = -1/5, and it intersects the m-axis at m = 50 and the s-axis at s = 10. The indifference curve for the utility level U(m*, s*) = 2√25 + 5 = 15 is a curve that is increasing at a decreasing rate. The optimal bundle (m*, s*) = (25, 5) is the point where the budget line is tangent to the indifference curve.

(c) If the price of milkshakes increases to $5, the new budget constraint is 100 = 5m + 10s. Solving this constraint for s gives s = 10 - 0.5m. Substituting this into the utility function gives U(m, s) = 2√m + 10 - 0.5m. Taking the derivative of this with respect to m and setting it equal to zero gives the new optimal consumption bundle (m*, s*):

m* = 16
s* = 2

(d) Again, I can't draw a graph here, but the new budget line has a slope of -5/10 = -1/2, and it intersects the m-axis at m = 20 and the s-axis at s = 10. The new optimal bundle (m*, s*) = (16, 2) is the point where the new budget line is tangent to the indifference curve for the utility level U(m*, s*) = 2√16 + 2 = 10.

(e) 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 milkshakes from 25 units to 16 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 milkshakes from 25 units to 16 units, minus the increase in consumption of sushi from 5 units to 2 units, which is a decrease of 7 units.

(f) The amount of additional income needed for Collin to achieve the initial level of utility is the amount that would allow him to purchase the initial bundle at the new prices. This is 5*25 + 10*5 = $175. The amount of additional income needed for Collin to purchase the initial bundle is the difference between this and his current income, which is $175 - $100 = $75. The ideal cost of living index is the ratio of the new cost of the initial bundle to the old cost, which is $175/$100 = 1.75.

Knowee AI · Accepted Answer

(a) Collin's utility function is U(m, s) = 2√m + s. His budget constraint is 100 = 2m + 10s. To find his optimal consumption bundle, we need to maximize his utility subject to his budget constraint.

Setting up the Lagrangian, we get: L = 2√m + s + λ(100 - 2m - 10s)

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

dL/dm = 1/√m - 2λ = 0
dL/ds = 1 - 10λ = 0
dL/dλ = 100 - 2m - 10s = 0

Solving these equations simultaneously gives the optimal consumption bundle (m*, s*):

m* = 25
s* = 5

(b) Unfortunately, I can't draw a graph here, but I can describe how to do it. On the horizontal axis, you have m (milkshakes) and on the vertical axis, you have s (sushi). The budget line is a straight line with a slope of -2/10 = -1/5, and it intersects the m-axis at m = 50 and the s-axis at s = 10. The indifference curve for the utility level U(m*, s*) = 2√25 + 5 = 15 is a curve that is increasing at a decreasing rate. The optimal bundle (m*, s*) = (25, 5) is the point where the budget line is tangent to the indifference curve.

(c) If the price of milkshakes increases to $5, the new budget constraint is 100 = 5m + 10s. Solving this constraint for s gives s = 10 - 0.5m. Substituting this into the utility function gives U(m, s) = 2√m + 10 - 0.5m. Taking the derivative of this with respect to m and setting it equal to zero gives the new optimal consumption bundle (m*, s*):

m* = 16
s* = 2

(d) Again, I can't draw a graph here, but the new budget line has a slope of -5/10 = -1/2, and it intersects the m-axis at m = 20 and the s-axis at s = 10. The new optimal bundle (m*, s*) = (16, 2) is the point where the new budget line is tangent to the indifference curve for the utility level U(m*, s*) = 2√16 + 2 = 10.

(e) 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 milkshakes from 25 units to 16 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 milkshakes from 25 units to 16 units, minus the increase in consumption of sushi from 5 units to 2 units, which is a decrease of 7 units.

(f) The amount of additional income needed for Collin to achieve the initial level of utility is the amount that would allow him to purchase the initial bundle at the new prices. This is 5*25 + 10*5 = $175. The amount of additional income needed for Collin to purchase the initial bundle is the difference between this and his current income, which is $175 - $100 = $75. The ideal cost of living index is the ratio of the new cost of the initial bundle to the old cost, which is $175/$100 = 1.75.

Question

Solution

Similar Questions

Upgrade your grade with Knowee