MICRO-Question 1Using calculus, please show the co

Solution

a

Definition of Pareto Efficiency: Pareto efficiency, or Pareto optimality, is a state of resource allocation where it is impossible to make any one individual better off without making at least one individual worse off

b

Condition for Pareto Efficiency: In a two-person, two-good economy, Pareto efficiency requires that the marginal rate of substitution (MRS) between the two goods be equal for both individuals. Mathematically, this can be expressed as:

MRS_{A} = MRS_{B}

where

MRS_{A}

and

MRS_{B}

are the marginal rates of substitution for individuals A and B, respectively

c

Using Calculus: To find the condition for Pareto efficiency using calculus, we need to set up the utility functions for both individuals. Let

U_A(x_1, x_2)

and

U_B(y_1, y_2)

be the utility functions for individuals A and B, respectively. The marginal rate of substitution for each individual is given by the ratio of the marginal utilities:

MRS_{A} = \frac{\partial U_A / \partial x_1}{\partial U_A / \partial x_2}

MRS_{B} = \frac{\partial U_B / \partial y_1}{\partial U_B / \partial y_2}

For Pareto efficiency, we set these equal:

\frac{\partial U_A / \partial x_1}{\partial U_A / \partial x_2} = \frac{\partial U_B / \partial y_1}{\partial U_B / \partial y_2}

d

Graphical Representation: To illustrate this condition graphically, we can use an Edgeworth Box. The Edgeworth Box is a graphical representation of the distribution of resources in a two-person, two-good economy. The contract curve within the Edgeworth Box shows all the points where the MRS of both individuals are equal, indicating Pareto efficient allocations

Answer

The condition for Pareto efficiency in a two-person, two-good economy is that the marginal rate of substitution (MRS) between the two goods must be equal for both individuals. This can be represented mathematically as

MRS_{A} = MRS_{B}

and graphically using an Edgeworth Box.

Key Concept

Pareto Efficiency

Explanation

Pareto efficiency occurs when no individual can be made better off without making another individual worse off. In a two-person, two-good economy, this requires that the marginal rate of substitution between the two goods be equal for both individuals.

Solution

a

Definition of Vickrey-Clarke-Groves (VCG) Mechanism: The VCG mechanism is a type of auction designed to incentivize truthful bidding. Each participant states their true value for the good, and the good is allocated to maximize total value. Each participant pays a "tax" based on the impact of their bid on the total value

b

Expected Profits Calculation: The expected profits for each owner can be calculated by considering the sum of the expected profits of the other 9 owners and subtracting the maximum expected profits that would be received by the other owners if the price were chosen to maximize their expected total profits

c

Price Determination: The price

p

at which the land will be offered to the developer is the one that maximizes the total profits of the owners. Given the probability function

G(p) = e^{-p / k}

, we need to find the price that maximizes the expected total profits

d

Probability Calculation: The probability that the developer will buy the land is given by the function

G(p) = e^{-p / k}

e

Expected Profit for Each Villager: Each villager's expected profit is calculated by dividing the total expected profit by the number of owners (10 in this case)

Answer

The price at which the land will be offered to the developer is the one that maximizes the total profits of the owners. The probability that the developer will buy the land is given by

G(p) = e^{-p / k}

. Each villager's expected profit is the total expected profit divided by 10.

Key Concept

Vickrey-Clarke-Groves (VCG) Mechanism

Explanation

The VCG mechanism ensures that each participant states their true value for the good, and the good is allocated to maximize total value. Each participant pays a "tax" based on the impact of their bid on the total value.

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