2. Basic Macroeconomic Model of General Equilibrium under Perfect Competition:Household faces decision about how much to consume and how much to work, it also hassome capital that it rents out,max log(C) + xlog (1 - Lh)C.Ls.t. C = wLh+rKhwhere C is consumption, L, is labour supplied by household (so 1 - L₁ is leisure), w is thewage, r is the rental rate on capital, and K is capital held/supplied by the household.¹There is a Firm with production function Y = LK}, where Y is production, and Lf islabour used in production (demanded), and K, is captial used in production. Firms solvethe profit maximization problemmax Y-wL-rKfY.Lj.Kjs.t. Y = LK-¹Since the Household does not like capital in an of itself it will always choose to rent out all the capital it possesses.1note that profits equal revenues (Y, as price is normalized to one) minus costs of the inputs;labour wLf (wages paid to labour) and capital rK, (cost of renting the capital).General equilibrium will be given by market clearance, here that means the labour marketclears L₁ = Lf and the capital market clears K₁ = Kf.(a) Solve firms maximization problem to show that wage equals the marginal product oflabour, ie. w = aL-¹K-a, and that the rental rate equals the marginal product ofcapital, ie. r = (1-a) LjK7a.(b) Solve households maximization problem to find that people will choose to work L1-XEK).(e) Show the following useful intermediate result: r/w = 1-ak(d) Imposing labour market clearance, L = L₁ = Lf, and capital market clearance, K =K; = Kh, show that in general equilibrium L = ax (Hint: combine previous resultson r/w and Lh.)(e) Show that in general equilibrium the other economic outcomes in this economy are givena-11-aby: w = a (x) K-ª, and r = (1-a) () Kª(f) Find equilibrium values for C using household budget constraint and Y using productionfunction. What do you notice about the values you have found for C and Y?(Hint: see (i) from 'Behind-the-Scenes' below.)(Remark: Congratulations, you have just solved for general equilibrium in a basic modelof perfect competition. Observe the feedback loop from wages, to labour supply (house-hold) and labour demand (firm), and back to wages.)

Utility function : U = log (C) + Xlog (1-Lh)Constraint : C = wLh + rKh Profit function : Y - wLf -…

Basic Macroeconomic Model of General Equilibrium under Perfect Competition: Household faces decision about how much to consume and how much to work, it also has some capital that it rents out, max log(C) + xlog (1-L₁) s.t. C = wLh+rKh where C is consumption, L, is labour supplied by household (so 1 - L₁ is leisure), w is the wage, r is the rental rate on capital, and K is capital held/supplied by the household.¹ There is a Firm with production function Y = LK}, where Y is production, and L; is labour used in production (demanded), and K; is captial used in production. Firms solve the profit maximization problem max Y-wLf-rKf Y.Lj.Kj s.t. Y = = LjK}-a nce the Household does not like capital in an of itself it will always choose to rent out all the capital it possesses.

Basic Macroeconomic Model of General Equilibrium under Perfect Competition: Household faces decision about how much to consume and how much to work, it also has some capital that it rents out, max log(C) + xlog (1-L₁) s.t. C = wLh+rKh where C is consumption, L, is labour supplied by household (so 1 - L₁ is leisure), w is the wage, r is the rental rate on capital, and K is capital held/supplied by the household.¹ There is a Firm with production function Y = LK}, where Y is production, and L; is labour used in production (demanded), and K; is captial used in production. Firms solve the profit maximization problem max Y-wLf-rKf Y.Lj.Kj s.t. Y = = LjK}-a nce the Household does not like capital in an of itself it will always choose to rent out all the capital it possesses.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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