note10_n3 - Econ303 A One Period Macroeconomic Model...

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Econ303 A One Period Macroeconomic Model 1 Competitive Equilibrium Revisited In the previous lecture we did not talk about the demand of labor and the determination of the wage rate. To explain the observed fact that there is a huge increase of the wage rate but no increase of labor input in the US after the world war II, we need to look at the supply side and the demand side jointly. In this lecture, we return back to the determination of a competitive equilibrium. In this particular case, we will model how the wage rate is determined and where does Mr. Crusoe gets his non-labor income. In order to do so, we add the production side of the economy. To simplify our analysis, we use a very simple production function. No capital is required in production; output is determined by labor input only. If Mr. Crusoe spends more time on picking bananas, he will get more bananas. Let Y denote output and L labor input. The production function is Y = F ( L ) Suppose that instead of eating what he can produce, Mr. Crusoe sets up a firm that hires himself and pays him based on how many hours he worked with a constant hourly wage. Mr. Crusoe also gets profits of the firm he owns. These profits are payments capital, i.e. any tools Mr. Crusoe has and rented to the firm. 1. Firm’s problem Prices in this economy are the price of output, normalized to be one, and the wage rate, w . Given prices, firm decides how much labor to hire and how much to produce to maximize profit, π : max L π = F ( L ) - w · L. The optimal choice satisfies the marginal condition, MP L = w. Labor Demand Function L d ( w ) From firm’s marginal condition we can get the demand function for labor. L d ( w ) is used to denote the labor demand function. L is used to represent a particular number. Once labor demand is determined, output or the supply of consumption good is also determined by Supply of good Y = F ( L ) Profits of the firm are given by π = F ( L ) - w · L 2. Household’s problem Households solves the following problem: max c,l u ( c, l )
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Econ303 A One Period Macroeconomic Model 2 subject to the budget constraint. c w · l + π Households’ optimal choice satisfies MRS = w c = w · l + π Labor Supply Function l s ( w, π ) Demand of good c The first equation is the marginal condition, which says that at the optimal work- consumption choice, the marginal rate of substitution between labor and consumption is equal to their price ratio, i.e. the wage rate. The second is the binding budget constraint. Here l s ( w, π ) is used to denote the labor supply function.
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