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Unformatted text preview: Econ 102 Winter 2012 Lecture Note 4 The Solow Model A Note on the Steady State We showed graphically and mathematically that the economy naturally goes to the steady state if there are no changes in exogenous variables. Since our focus is on longrun economic growth, we are going to assume that much of the growth that we see over long periods of time occur when the model has reached some sort of steady state. Sources of growth Remember that the sources of growth in the Solow model are all exogenous. In the current version of the model, these exogenous variables include • s the savings rate • A the level of technology • δ the depreciation rate • α the share of capital We can see the effects of changing these variables on the formula for steadystate capital: k ss = ( sA δ ) 1 1 α Increases in s , A and α will increase the steady state level of percapita capital (see the solution section below for more on this), while increases in δ will lower the steady state level of percapita capital. Population Growth The first exogenous source of growth we will add is a growing population. Throughout history, economic growth has been associated with population growth. Economists have proposed that population growth might be responsible for economic growth. In order to assess this claim in the framework of the Solow model, we will add population growth and see if the results of the model match our growth facts. Specifically, we will assume that the population in our model grows at a constant rate n (which is given): L t +1 = L t (1 + n ) 1 And in continuous time (we will use this to determine growth rates later), ∂ log L t ∂t = n Going back to our original Solow model equations: Y t = AK α t L 1 α t I t = sY t K t +1 = K t + I t − δK t Just as in the case without population growth, we want to express the model in percapita terms. Dividing the equations by L t , the first two equations are unchanged: y t = Ak α t i t = sy t Dividing the third by L t , K t +1 L t = k t + i t − δk t We have to be careful, because K t +1 L t is not the same as k t +1 , since k t +1 ≡ K t +1 L t +1 and L t +1 ̸ = L t when there is population growth. If we use the rule for population growth, we can show that: k t +1 = K t +1 L t +1 = K t +1 (1 + n ) L t Replacing this into the third equation, we get that the percapita version of the model is: y t = Ak α t i t = sy t (1 + n )( k t +1 − k t ) = i t − ( n + δ ) k t 2 Steady State Remember that the definition of steady state is where k t +1 − k t =: k ss : (1 + n )( k ss − k ss ) = i ss − ( n + δ ) k ss = 0 and the equation for i ss is: i ss = sAk α ss So at k ss , sAk α ss = ( n + δ ) k ss Just as in the previous set of notes, we can solve for k ss to see how the exogenous variables affect the steady state: k ss = ( sA n + δ ) 1 1 α Notice that the difference between steady state capital in an economy with population growth and one without is that the population growth rate n enters the condition. Interestingly,enters the condition....
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 Spring '11
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 Steady State, Kt Lt

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