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, π‘₯π‘₯1 and π‘₯π‘₯2. The utilities that both A and B get from the consumption of these goods are given by the functions;
π‘ˆπ‘ˆπ΄π΄(π‘₯π‘₯1𝐴𝐴,π‘₯π‘₯2𝐴𝐴) = (π‘₯π‘₯1𝐴𝐴)1/3. (π‘₯π‘₯2𝐴𝐴)2/3 and π‘ˆπ‘ˆπ΅π΅(π‘₯π‘₯1𝐡𝐡,π‘₯π‘₯2𝐡𝐡) = (π‘₯π‘₯1𝐡𝐡)1/3. (π‘₯π‘₯2𝐡𝐡)2/3, respectively. Before the exchange starts between the consumers, A has 9 units of π‘₯π‘₯1 and 6 units of π‘₯π‘₯2, that is (πœ”πœ”1𝐴𝐴, πœ”πœ”2𝐴𝐴) = (9, 6). B has 18 units of π‘₯π‘₯1 and 3 units of π‘₯π‘₯2, that is (πœ”πœ”1𝐡𝐡, πœ”πœ”2𝐡𝐡) = (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).
π‘€π‘€π‘Žπ‘Žπ‘₯π‘₯ π‘ˆπ‘ˆπ΄π΄(𝑋𝑋𝐴𝐴,π‘Œπ‘Œπ΄π΄) 𝑠𝑠𝑒𝑒𝑏𝑏𝑗𝑗𝑒𝑒𝑐𝑐𝑑𝑑 π‘‘π‘‘π‘œπ‘œ
π‘ˆπ‘ˆπ΅π΅(𝑋𝑋𝐡𝐡,π‘Œπ‘Œπ΅π΅) = 𝑍𝑍 (B’s utility is held at some arbitrary level Z)
𝑋𝑋1(𝐾𝐾1𝑋𝑋,𝐿𝐿1𝑋𝑋) + 𝑋𝑋2(𝐾𝐾2𝑋𝑋,𝐿𝐿2𝑋𝑋) = 𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐡𝐡 (Total consumption of X is equal to the amount produced)
π‘Œπ‘Œ1(𝐾𝐾1π‘Œπ‘Œ,𝐿𝐿1π‘Œπ‘Œ) + π‘Œπ‘Œ2(𝐾𝐾2π‘Œπ‘Œ,𝐿𝐿2π‘Œπ‘Œ) = π‘Œπ‘Œπ΄π΄ + π‘Œπ‘Œπ΅π΅ (Total consumption of Y is equal to the amount produced)
𝐾𝐾𝑇𝑇 = 𝐾𝐾1𝑋𝑋 + 𝐾𝐾2𝑋𝑋 + 𝐾𝐾1π‘Œπ‘Œ + 𝐾𝐾2π‘Œπ‘Œ (The sum of the capital input across all firms is equal to the economy’s respective endowment, KT)
𝐿𝐿𝑇𝑇 = 𝐿𝐿1𝑋𝑋 + 𝐿𝐿2𝑋𝑋 + 𝐿𝐿1π‘Œπ‘Œ + 𝐿𝐿2π‘Œπ‘Œ (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 βˆ’ 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
π‘Šπ‘Š = π‘€π‘€π΄π΄π‘ˆπ‘ˆπ΄π΄(𝑋𝑋𝐴𝐴) + π‘€π‘€π΅π΅π‘ˆπ‘ˆπ΅π΅(𝑋𝑋𝐡𝐡) where π‘ˆπ‘ˆπ΄π΄(𝑋𝑋𝐴𝐴) is the utility function of person A from consumption of X and π‘ˆπ‘ˆπ΅π΅(𝑋𝑋𝐡𝐡) is the utility of person B from consumption of X. Total consumption in the society cannot exceed the endowment of X, that is 𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐡𝐡 ≀ 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.