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# ln3-revised - Econ 102 Winter 2012 Lecture Note 3 The Solow...

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Econ 102 Winter 2012 Lecture Note 3 The Solow Growth Model (with no growth) The Solow model is a simple model of economic growth, with one good that is produced by a Cobb-Douglas production technology. The good is used for either consumption ( C ) or investment ( I ): Y = C + I ; there is no government and no trade with other countries. Investment is converted to capital ( K ), but not immedi- ately. We will assume that the markets for consumption, capital and labor are all perfectly competitive (ﬁrms are price takers). In addition, we assume that everyone works - there is no diﬀerence between L t and the population at time t . One of the key assumptions we make is that the savings rate of the country is given fraction s and not chosen. This means a fraction s of output will be devoted to investment in each period. In addition, capital depreciates at a constant rate δ . This means that capital, if left alone, disappears at a rate δ . In order to sustain the same amount of capital, the economy needs to replenish its capital stock ( K t ) with new investment. The Solow Model, Mathematically The above assumptions are consistent with an economy described by these equations: Y t = F ( K t ,L t ) = AK α t L 1 - α t (1) I t = sY t (2) K t +1 = K t + I t - δK t (3) (1) Output is produced by a Cobb-Douglas production function (2) Investment is a constant fraction of output (3) Capital depreciates at rate δ and accumulates with investment The values for A , α , δ , and s are all given in this model. Variables and parameters that are given from outside the model are called exogenous (Greek for “given from outside”), while variables like Y t , K t and I t , which are determined within the model, are called endogenous . 1

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Per-capita Version In order to simplify our analysis, we can set up a per capita version of the model. Since F ( K,L ) is constant returns to scale, Y t = F ( K t ,L t ) = Y t L t = F ( K t /L t , 1) Whenever we use lower-case variables, we will mean per-capita. So Y t is total output while y t will stand for per-capita output. Mathematically, y t Y t L t , k t K t L t , i t = I t L t , etc. For our production function, this means: y t = F ( k t , 1) = Ak α t (1) 1 - α Since this looks a bit weird, we’ll use a lower-case f ( k t ) to mean F ( k t , 1) y t = f ( k t ) = Ak α t In order to discuss growth, ﬁrst we examine what happens in the model over time. Our capital accumulation equation tells us that capital in the next period comes from capital in the current period, plus investment, minus depreciation. k
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ln3-revised - Econ 102 Winter 2012 Lecture Note 3 The Solow...

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