Ch7_8_handout - Chapters 7 and 8 Solow Growth Model Basics...

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Chapters 7 and 8 – Solow Growth Model Basics The Solow growth model breaks the growth of economies down into basics. It starts with our production function Y = F ( K, L ) and puts in per-worker terms. Y L = F ( K L , L L ) y = f ( k ) (1) where k is the amount of capital per worker and y is the amount of output per worker. The slope of this function measures the change in output per worker due to a one unit increase in capital per worker which, as we saw from chapter 3, is equal to the MPK . Thus the slope of (1) is f ( k ) = MPK . Due to the decreasing marginal productivity of capital, this is decreasing in y , making f ( k ) a concave function. Individuals consume whatever they do not save, where s is the savings rate, somewhere be- tween 0 and 1 c = (1 - s ) y (2) All output is either allocated to consumption or investment. y = c + i (3) By combining equations 2 and 3, we can show that i = sy . 1 The Steady State What changes k ? For now, we’ll look at depreciation and population growth. Population increase (denoted as a percentage by n ) doesnt actually affect the amount of capital ( K ) in our economy. Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2008
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What it does do, however, is decrease the amount of capital per worker ( k ). Depreciation (denoted by δ ) is the rate at which capital wears out. These two factors combined are eating away at our capital per worker on a regular basis. In order to retain an unchanging level of capital per worker k over time, we have to invest enough to create new capital to offset this loss over time. Thus, to maintain a “steady state” where capital per worker is constant over time, we must have that; Δ k = sf ( k ) investment in new capital - ( δ + n ) k “loss” in capital = 0 sf ( k * ) = ( δ + n ) k * where * indicates steady state values. Note how this shows that as our capital per worker k gets larger, larger amounts of investment are required to maintain Δ k = 0 . The economy will always work itself to a steady state point. If the rate of capital replenishment is greater than the loss due to depreciation and population growth ( sf ( k ) > ( δ + n ) k ), then the capital stock will grow. If the
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