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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
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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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 has some capital that it rents out, max log(C) + xlog (1 - Lh) C.L 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 Lf is labour used in production (demanded), and K, is captial used in production. Firms solve the profit maximization problem max Y-wL-rKf Y.Lj.Kj s.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. 1 note 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 market clears L₁ = Lf and the capital market clears K₁ = Kf. (a) Solve firms maximization problem to show that wage equals the marginal product of labour, ie. w = aL-¹K-a, and that the rental rate equals the marginal product of capital, ie. r = (1-a) LjK7a. (b) Solve households maximization problem to find that people will choose to work L 1-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 results on r/w and Lh.) (e) Show that in general equilibrium the other economic outcomes in this economy are given a-1 1-a by: w = a (x) K-ª, and r = (1-a) () Kª (f) Find equilibrium values for C using household budget constraint and Y using production function. 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 model of perfect competition. Observe the feedback loop from wages, to labour supply (house- hold) and labour demand (firm), and back to wages.)
Transcribed Image Text: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 has some capital that it rents out, max log(C) + xlog (1 - Lh) C.L 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 Lf is labour used in production (demanded), and K, is captial used in production. Firms solve the profit maximization problem max Y-wL-rKf Y.Lj.Kj s.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. 1 note 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 market clears L₁ = Lf and the capital market clears K₁ = Kf. (a) Solve firms maximization problem to show that wage equals the marginal product of labour, ie. w = aL-¹K-a, and that the rental rate equals the marginal product of capital, ie. r = (1-a) LjK7a. (b) Solve households maximization problem to find that people will choose to work L 1-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 results on r/w and Lh.) (e) Show that in general equilibrium the other economic outcomes in this economy are given a-1 1-a by: w = a (x) K-ª, and r = (1-a) () Kª (f) Find equilibrium values for C using household budget constraint and Y using production function. 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 model of perfect competition. Observe the feedback loop from wages, to labour supply (house- hold) and labour demand (firm), and back to wages.)
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