Energy and Environmental Economics

Posted: August 25th, 2021

Student’s Name

Professor’s Name

Course

Date

Energy and Environmental Economics

Question 1

1. Edgeworth Box (2 x 2 model)

The two commodities include 𝑥1 and 𝑥2 and consumers are A and B. Endowment refers to goods that consumers initially possess, in this case, for A aris units of 𝑥1 and 6 units of 𝑥2 that is, (𝜔1𝐵, 𝜔2𝐵) = (9, 6).while B has 18 units of 𝑥1 and 3 units of 𝑥2, that is,  (𝜔1𝐵, 𝜔2𝐵) = (18, 3).

• What is the price of second good, p2 if price of first good is p1 = $1

Income, I = endowment, w1 *price, p1 + endowment, w2 * price, p2

But; p1 *commodity, x1 + p2*commodity, x2 should equal the price.

Hence;
 for consumer A

d

but w1A = 9, and p1 =$1 hence at equilibrium, x1=x2 => $9

Question 2: ConditionsCharacterizing Allocative Efficiency

Allocative efficiency is concerned about how the resources are allocated particularly about societal preferences. Every commodity is produced to a level at which the last unit utilized in production offers marginal benefit to the consumers equivalent to marginal production cost. For instance, the point of allocative efficiency is indicated where the price is equivalent to the marginal cost. Based on this background, allocated efficiency for the three components can be derived as follows;

The constrained maximizing problem;

  subject to

Where X and Y are commodities and A and B are consumers.

Using first-order Lagrangian conditions,

1. Efficiency in consumption allocation

First-order conditions for consumption of individuals A and B is obtained by setting the Langrangian consumption equation:

Max. The utility of A for commodity X

Maximum utility for A is attained when:

Max. The utility of B for commodity Y

The maximum utility of B is attained when

1. Efficiency in Production Allocation

Production efficiency is attained when based on the level of satisfaction a consumer gets from the goods produced. Thus:

 subject to

For efficient production:

Langrangian equation for production:

Efficiency allocation is attained when:

1. Efficiency in Product Mix

Using eq. 1, efficiency in product mix will be;

Question 3

Honey farm production cost => C_H (H, A) =

Orchard farm production cost => C_A (H, A) =

Where H and A are pounds of honey and apple production.

The market price of honey,P_H = $7 and for apples, P_A= $5

1. Amount of Apple Produced by Orchard Farm If Apple Market Is Competitive

For a competitive market, production should be efficient, implying that the orchard company has to minimize costs of production apples while ensuring maximum production. Thus,

Minimize C_A (H, A) => = 1        eq. 1

• Amount of Apple Produced If Orchard Farm Acquires Honey Farm And Produces Honey

Acquiring a honey farm implies incurring the costs of production in the two commodities. Thus, to minimize costs and attain efficiency in production, the following equation applies;

Minimize C_A (H, A) =

• Comparing the two quantities in (a) and (b), it is shown that the production of apples when orchard farm acquires honey farm is higher than when it is operating alone. The reason is that the farm gains economies of scale and it can meet its costs efficiently once it acquires a honey farm. There is also reduced competition and hence increase in market share.

Question 4: Given Social Welfare Function as:

Where U^A(X^A) is the utility function of a person A form consumption of X and U^B(X^B) is the utility function of a person B from consumption of X. Total consumption cannot exceed endowment of X, given by X^A + X^B ≤𝑋 ̅. 

Unequal distribution of goods can occur at welfare maximum if weights attached to individual utilities are unequal and/or when individuals have different utility functions.

Demonstration

To start with, consumers should choose X^A and  X^B to maximize utility represented as;

Subject to constraints;

X^A + X^B ≤𝑋̅

Langrangian equation to formulate the problem:

X^A – X^B )     eq. 1

First-order conditions:

looking at eq. 1 and eq. 2, it implies that,

   eq. 4

When both consumers are using similar utility function such that;

U^A(X^A) = U^B(X^B) = U (X), then utility maximization condition becomes;

In this case, W_A unequal to W_B condition becomes available if X^A is unequal to X^B that essentially implies there is no equal distribution of goods. If there is a difference in utility function but  W_A equal to W_B, dividing both sides by W_A gives;

   but only true if X^A is unequal to X^B.