Question

3. Consider two consumers (Ann and Bob) who trade apples and broccoli. Before trading, Ann owns 8 pounds of apples and 2 pounds of broccoli. Bob, on the other hand, owns 2 pounds of apples and 8 pounds of broccoli. Ann's preferences are represented by the utility function usa,bA) = 20A + 5b4. Bob's preferences are represented by the utility function ubla,bB) = abs. (a) Draw an Edgeworth box representing this economy. Clearly label the initial endow- ment and highlight the set of feasible allocations that Pareto dominates it (these are the feasible allocations that can be reached through voluntary trade). (b) Find the set of feasible allocations that are Pareto efficient. Explain how you obtained it. Describe the set mathematically and mark it on the Edgeworth box. (c) Compute the competitive equilibrium prices and the associated allocations. Are oqui librium allocations Pareto efficient?

Answer 1

consumers, xit A&B be two apples a brocolli & b. Endowmcuts UAl AA , BA) = 2aA +56 A. UB (ab, bB) = abbo. (812) (278) brocolli the black wies are Ic of A which are with slope 2/s. The green curues are Inicial endowment is Pauit w a { (812); (2,8) } . substitutes IC of B Any allocation on the It passing via w l for Pareto superius allocation is an allocation where - no body is worse off a atleast one indiuidug strictly better off s es pareto superior to s' il ai (s)>, uils) #set tien ails) > uics I for alleast onese CEN

Date B DELTA PO NO 100 8 10 Any allocation inside ne avea enclosed by ICA & ICB is have to superior to w as alleast I individuel is strictly better off & no one is worse off. 6 Pareto effiuent allocation is one where you cannuot find a pareto silperior allocation & it must be feasible. lo The fet points A,B,C all are PE and There is no pareto superior allocation to these point - here & MRSA = MRS B - the two ICs are tangent to each other. (1 to check if they are PE Take any other point x a to show it is not parreto efficient! If we move from X to c the IC of B is same so it is has the same utility but Iho we are on a higher IC for A. Hence A is strictly better off while B is no worse off

c is rence .. pareto superior to X cant be PE. OA OB are pe ie . the points on contract curve there is no p.S allocation to mese. Mathematically & the set of points where MRSA=MRSB. are PE 2= bb 5 AB. Lag=5bbl contract 2 1 10-2A) = 5 (10-ba) 20-204 = 5o- sba 15bA - 2a4 = 301 curre

DELTA Pg No Competitie egy CLE): Marsheliau demands For A u= 204 + sba be a baş CA, bon od Milo Poc 7 Pa 22 Po 5 of Mo if a L 23 C. if a 2/5 аа | Раа. + Pb bл 4 м, у ра – 2/s Т 6 For B: UB= abbo. at eg in Slope of IC = slope of BL (you can also use MRS = palpb. At in Kang rangean) bB = Pa a) bB = ab Pa AB + Pb. A Pb ,- Puttup it in B.L. S pa. AB + Pbl abba l-M2 AB - M2 , bB = M2 altls/ alati 2Pb.

For CE , he markets must clear it - QAtaB = w êt war = 10 ba+bos = W6A + ? = 10. checking for market clearance assume price vector - PE [Pi] as we take bas numeraine. li) of P( relative price) L215 se men antaB=10. en Mit M2 =10 o per 2Pa Mi = pawåt Pb WGA = 8pa +2. M2 = pa waB +PG w 3 =28a+8. LHS) Now, 2(8 pest2) + 2Pet8 apa lopaty +220 +8 2le - 12 Pa + 12 = - 2P 6 Perb pe Rhs 10 - - 6Path = 10 - upe = 6 Pa- 3/2 Pa is not un han 2/5 not possible-

at P7215 aAtaB =10 ot M2 =10 2P = 2P+8 =10 2P = 180=8 P = 419.7215 . competitive price is P44/9,1) Pa=4 Pb=9 . or

For A. Alw cations: MI 8P+2 AA=0 bA= 22 864/9)+2 = 3270 2 18 = 2.8. = for Bi AB= M2=2P+8 - 2P 2P = 2(4/9)+8 21419) 8 +72 8 = 10 do = M2 = 214/9+8 = 8 +72 a 2 = 4.44 18 Hence : allocati ens : Ş (012:8), 610, 4.44)' to check PE see if it satisfies contract warue equ 5ba- 2aA=30 5(2.8) -0 =30 NO (PH It is not in the sentract connue ACE is not PET