EC571 ENERGY AND ENVIRONMENTAL ECONOMICS FALL 19
Homework Problem Set due October 3
First and Last Name: _____________Student ID#: _____________ Date: _________
Due Date: At the beginning of the class on Thursday, October 3rd, 2019. Submit the printed and stapled copy to me. Scanned and e-mailed copies are not accepted as submissions.
1. Please provide your solution, explanation and drawing for every single question.
2. Please write clearly. If your answers are not legible they will have zero points.
3. Graphs: Please label the axes, lines, curves and areas correctly. Graphs with no labels will not receive any points.
Q.1. (25 Points) In a 2×2 exchange economy, A and B consume two goods, 𝑥𝑥1 and 𝑥𝑥2. The utilities that both A and B get from the consumption of these goods are given by the functions;
𝑈𝑈𝐴𝐴(𝑥𝑥1𝐴𝐴,𝑥𝑥2𝐴𝐴) = (𝑥𝑥1𝐴𝐴)1/3. (𝑥𝑥2𝐴𝐴)2/3 and 𝑈𝑈𝐵𝐵(𝑥𝑥1𝐵𝐵,𝑥𝑥2𝐵𝐵) = (𝑥𝑥1𝐵𝐵)1/3. (𝑥𝑥2𝐵𝐵)2/3, respectively. Before the exchange starts between the consumers, A has 9 units of 𝑥𝑥1 and 6 units of 𝑥𝑥2, that is (𝜔𝜔1𝐴𝐴, 𝜔𝜔2𝐴𝐴) = (9, 6). B has 18 units of 𝑥𝑥1 and 3 units of 𝑥𝑥2, that is (𝜔𝜔1𝐵𝐵, 𝜔𝜔2𝐵𝐵) = (18, 3).
a. Draw an Edgeworth Box of this economy. Clearly mark (1) the axes, (2) size of the box and mark (3) the endowment point.
b. If you pick the first good as a numeraire so that 𝑝𝑝1 = $1, what is the price of the second good, 𝑝𝑝2 = $?, in the equilibrium?
Q.2. (35 Points) Consider an economy with two individuals (A and B) each consuming two commodities (X and Y), where each commodity is produced by an industry comprising of two firms (1 and 2), each of which uses two inputs – capital and labor (K and L). Derive the conditions characterizing allocative efficiency ((i) efficiency in consumption, (ii) efficiency in production, and (iii) efficiency in product mix) by considering the following constrained maximization problem. (Hint: One can use the first order conditions of the Lagrangian function to derive the efficiency conditions).
𝑀𝑀𝑎𝑎𝑥𝑥 𝑈𝑈𝐴𝐴(𝑋𝑋𝐴𝐴,𝑌𝑌𝐴𝐴) 𝑠𝑠𝑢𝑢𝑏𝑏𝑗𝑗𝑒𝑒𝑐𝑐𝑡𝑡 𝑡𝑡𝑜𝑜
𝑈𝑈𝐵𝐵(𝑋𝑋𝐵𝐵,𝑌𝑌𝐵𝐵) = 𝑍𝑍 (B’s utility is held at some arbitrary level Z)
𝑋𝑋1(𝐾𝐾1𝑋𝑋,𝐿𝐿1𝑋𝑋) + 𝑋𝑋2(𝐾𝐾2𝑋𝑋,𝐿𝐿2𝑋𝑋) = 𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 (Total consumption of X is equal to the amount produced)
𝑌𝑌1(𝐾𝐾1𝑌𝑌,𝐿𝐿1𝑌𝑌) + 𝑌𝑌2(𝐾𝐾2𝑌𝑌,𝐿𝐿2𝑌𝑌) = 𝑌𝑌𝐴𝐴 + 𝑌𝑌𝐵𝐵 (Total consumption of Y is equal to the amount produced)
𝐾𝐾𝑇𝑇 = 𝐾𝐾1𝑋𝑋 + 𝐾𝐾2𝑋𝑋 + 𝐾𝐾1𝑌𝑌 + 𝐾𝐾2𝑌𝑌 (The sum of the capital input across all firms is equal to the economy’s respective endowment, KT)
𝐿𝐿𝑇𝑇 = 𝐿𝐿1𝑋𝑋 + 𝐿𝐿2𝑋𝑋 + 𝐿𝐿1𝑌𝑌 + 𝐿𝐿2𝑌𝑌 (The sum of the labor input across all firms is equal to the economy’s respective endowment, LT)
Q.3. (20 Points) There are two adjacent farms; one produces apples and the other produces honey. The farm keeping the bees and producing honey has production cost of CH(H,A) = H2/100 − 3A and the orchard farm has production cost of CA(H,A) = A2/100 where H and A are the pounds of honey and apple production, respectively. The market price of honey, pH, is $7 and the market price of apples pA is $5.
a. How much apple does the orchard farm produce, if the apple market is competitive?
b. How much apple does the orchard farm produce, if it acquires the honey farm and produces both honey and apple?
c. Compare the apple production in part a with the apple production in part b. Please provide an explanation for why the apple productions are in different amounts in a and b. Why is it higher or lower when the orchard farm owns the honey farm?
Q.4. (20 Points) The welfare function of a society is given as
𝑊𝑊 = 𝑤𝑤𝐴𝐴𝑈𝑈𝐴𝐴(𝑋𝑋𝐴𝐴) + 𝑤𝑤𝐵𝐵𝑈𝑈𝐵𝐵(𝑋𝑋𝐵𝐵) where 𝑈𝑈𝐴𝐴(𝑋𝑋𝐴𝐴) is the utility function of person A from consumption of X and 𝑈𝑈𝐵𝐵(𝑋𝑋𝐵𝐵) is the utility of person B from consumption of X. Total consumption in the society cannot exceed the endowment of X, that is 𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ≤ 𝑋𝑋̅. Demonstrate that an unequal distribution of goods at a welfare maximum may occur (a) when the weights attached to individual utilities are not equal, and/or (b) when individuals have different utility functions.