Output√w2. Consider a one-consumer one-firm economy in which u(x) = x₂ + √√q and f(2)=√z.(a) Derive the Pareto set of this economy.Solution. Pareto-efficient allocations are solutions to the problem₂+√√q subject to √w- x₂ = xq,which is illustrated in Figure 3. Any solution to this problem in which , x > 0 mustsolve the first-order condition= f'(wx), or 2₁/x =max0≤x≤w,xq 20au(x*)Əxz_u(x*)ƏxqBy substituting = √w - x, we have that x = (¹/2)¹/³ and x = w − (1/2)³/³.Any solution to the Pareto problem with x = 0 would have to be such thatəFigure 5: No corner solution with x = 0Əxz— *²)=0²u(x*)x=0which, using that = √w in this case, reduces to w≤ (1/2) 8/³. In words, if theconsumer's endowment of input is small, then the Pareto-efficient allocations havethe firm use up all of the input to produce output, leaving the consumer with noconsumption of input. This is illustrated in Figure 4.Any solution to the Pareto problem with x = 0 would have to be such that≤ f(w-x)-u(x*)u(x²)x=0which, using that x = win this case, reduces to 0 ≥ 1/2√w-xx=0~∞. Thiscondition never holds, so that x = 0 is never optimal. In words, if the firm isproducing no output, then its marginal productivity is arbitrarily high, whereas ifthe consumer is not consuming any output, then her marginal utility from outputis also arbitrarily high. Therefore, it would benefit the consumer and be feasible toproduce some output in this case, so that no-production is Pareto-dominated. This isillustrated in Figure 5.→ Input12√w-x− x)\z;=0"> f'(w-:Therefore, the Pareto set of this economy contains only allocations x = (w-(1/2)/³, (1/2)1/³)and * = ((1/2), (1/2)4³) if w > (1/2), and it contains only allocations * = (0,√)and * = (w, √w) if w ≤ (1/2)*/³.(b) Fix some Pareto-efficient allocations from (a) and derive a competitive equilibrium forthis economy that generates these allocations.Solution. Suppose that w > (1/2), so that the Pareto-efficient allocations from(a) have x > 0. The slope of the consumer's indifference curve through thePareto-efficient allocations is 2√ = 2. Therefore, the prices for the equilibriumthat we are constructing must be p* = (1, (1/2)). Given prices p*, we know thatthe firm's optimal production plan is o* = (1/4(Pi/p:)², 1/2(Pi/p:)) = ((¹/2)/³, (1/2)1/³),with corresponding profits II* = P²/4p = (1/2). Therefore, prices p*, allocationsx = (w (1/2)/³, (1/2)4³) and * = ((1/2), (1/2)4³), along with profits II* = (1/2) 8/3form a competitive equilibrium. This is illustrated in Figure 6.0Output√wFigure 3: Pareto problem for Question 2OutputNg=√125-1/₂₂WFigure 4: Corner solution with z = 0vài ngکلاInputInput

The answer provided below has been developed in a clear step by step manner.Step: 1Introduction To…

2. Consider a one-consumer one-firm economy in which u(x) = x₂ + √/Tq and ƒ(z) = √√z. (a) Derive the Pareto set of this economy. Solution. Pareto-efficient allocations are solutions to the problem max x₂ +√√q subject to √w- x₂ = xq, 0<= 0 must solve the first-order condition a ərz ə Ər u(x*) au(x*) diq By substituting = √w - r, we have that x = (¹/2)¹/³ and x = w − (1/2)8/³. Any solution to the Pareto problem with x = 0 would have to be such that -u(x*) u(x*) z=0 1 2√w-x = f'(w — x²), or 2√x = 2√/w < f'(w − x ²) |_z_0² =0'

2. Consider a one-consumer one-firm economy in which u(x) = x₂ + √/Tq and ƒ(z) = √√z. (a) Derive the Pareto set of this economy. Solution. Pareto-efficient allocations are solutions to the problem max x₂ +√√q subject to √w- x₂ = xq, 0<= 0 must solve the first-order condition a ərz ə Ər u(x) au(x) diq By substituting = √w - r, we have that x = (¹/2)¹/³ and x = w − (1/2)8/³. Any solution to the Pareto problem with x = 0 would have to be such that -u(x) u(x) z=0 1 2√w-x = f'(w — x²), or 2√x = 2√/w < f'(w − x ²) |_z_0² =0'

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Suppose the household’s preferences are given the following forms: u(c)=log(c), v(l)=log(l), that ß=2/3, and that the production functions are given by: F1(L)=L1/2 and F2(K)=K1/2. a) What are the equilibrium conditions? Derive them. b) What are the quilibrium quantities?
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