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NOTES ON SOLOW GROWTH MODEL L t = Labor Force at beginning of year t K t = Capital Stock at beginning of year t C t = Consumption over year t I t = Investment over year t Y t = Output over year t k t = K t /L t = capital per worker c t = C t /L t = consumption per worker (“standard of living”) i t = I t /L t = investment per worker y t = Y t /L t = output per worker Distinguish between LR economic growth versus fluctuations around the trend Assumptions : 1. NX = 0 and G = 0 b Y = C + I 2. Labor force grows at the same rate as the population (let n = population growth rate) 3. Cobb-Douglas production function: Y t = A*K t α L t 1- α . PER-WORKER PRODUCTION FUNCTION To convert this production function into the “per-worker production function”, divide both sides by L t : (Y t /L t )= (A*K t α L t 1- α )/L t = A*K t α L t - α = (A) (K t /L t ) α b y = Ak α . This is the per-worker production function. For convenience, I will be dropping the time subscript for the per-worker variables (with a few exceptions) and use lower-case letters. y ( output per worker) y = Ak α Δ y Δ k k (capital per worker) Slope = ( Δ y/ Δ k) = MPK Notice that the slope gets flatter and flatter. This is due to diminishing marginal returns to capital (i.e., MPK falls as k rises, other things constant )
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The conclusion of the Solow Growth Model is that in the absence of technology changes, the economy ends up in a steady-state. A steady-state is, in this context, a situation in which k,i,c, and y are constant. NOTE: It is the ratio that is constant NOT the level . To see this, we need to first study investment in a steady-state. 1. I t = δ K t + (K t+1 – K t ) where δ is the depreciation rate of capital Investment to Net investment replace worn out capital Gross investment (the lefthand side—I t ) is used to replace worn out capital (the term δ K t ) and to add to the capital stock. The addition to the capital stock is called net investment and is given by the term (K t+1 – K t ) . Question : What is net investment (the last term on the right-hand side of the above equation) in a steady state ? The answer is nK t . To understand this, consider the following three points: a. k = K t /L t . b. k is constant in a steady state. c. L t grows at annual rate of n. As you can see, if the denominator grows at rate n, and the ratio k is constant, then the numerator must also grow at rate n. ------------------------------------------------------------------------------------------------------------------ NOTE —Perhaps a better way to understand this last point is as follows. 1. L t+t = (1+n)L t by the assumption that the labor force grows at rate n. 2. K t+1 = (1+g)K t with g = growth rate of capital. If we divide (2) by (1) we get: (K t+1 / L t+1 ) = [(1+g)/(1+n)] (K t / L t ) k t+1 = [(1+g)/(1+n)] (k t ) In a steady state, k t+1 = k t which can only be true if g = n. And so
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