note4_n3 - Econ303 A Simple Model of GDP Determination We...

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Econ303 A Simple Model of GDP Determination 1 We make the following assumptions Assumption 1 All countries use the same technology, Y = F ( K, L ) . Assumption 2 The supplies of labor and capital are exogenously given. Let L s denote the supply of labor and K s the supply of capital. Then L s = ¯ L and K s = ¯ K . Both ¯ L and ¯ K are some numbers. Firms in a competitive market maximize profits. Their optimal demand for capital and labor are K d and L d , which is determined by marginal conditions : MP K = ˆ r MP L = w At the equilibrium on the labor and capital markets, demands equal supplies. These are called market clearing conditions L d = L s K d = K s Definition 1 A competitive equilibrium consists of two sets of numbers: quantities ( Y, K d , L d , K s , L s ) and prices r, w ) . They satisfies the following requirements: 1. Given prices r, w ) , quantities ( Y, K d , L d ) maximize firms profits; 2. r, w ) are prices that clear markets. In another word, the equilibrium is determined by both marginal conditions and market clearing conditions. Mathematically, ( Y, L d , K d , L s , K s , w, ˆ r ) satisfies the following sets of equations MP K ( K = K d , L = L d ) = ˆ r MP L ( K = K d , L = L d ) = w L d = L s = ¯ L K d = K s = ¯ K Y = F ( ¯ K, ¯ L ) Below is how we find a competitive equilibrium for an economy: 1. The equilibrium quantities are given by the exogenous supplies of factors. (We use market clearing condition here.) At equilibrium K d = K s = ¯ K , L d = L s = ¯ L and Y = F ( ¯ K, ¯ L ).
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Econ303 A Simple Model of GDP Determination 2 2. The equilibrium prices are given by the marginal conditions. The rental rate of capital equals the marginal product of capital and the wage rate equals the marginal product of labor. Both marginal product are evaluated at the equilibrium capital and labor. That is, ˆ r = MP K ( ¯ K, ¯ L ) , w = MP L ( ¯ K, ¯ L ) . Point: In this simple model, GDP, wage rate and rental rate of capital in an economy are determined by the total amount of labor and capital supplied. example Suppose that the production function is Y = K 0 . 5 L 0 . 5 and the labor supply in this economy is L s = 0 . 5 and capital supply K s = 0 . 5. How do we find the equilibrium for this economy? The marginal product of capital, MP K = 0 . 5 K - 0 . 5 L 0 . 5 and the marginal product of labor, MP L = 0 . 5 K 0 . 5 L - 0 . 5 . Both are functions of capital and labor input.
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