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ln5-revised - Econ 102 Winter 2012 Lecture Note 5 Solow...

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Econ 102 Winter 2012 Lecture Note 5 - Solow + Intertemporal Consumption The Golden Rule The last concept of the Solow Model is the golden rule. 1 So far we have blindly taken all of the variables in the model as given. This might be troubling, especially the assumption that savings is a fixed proportion of output. After all, every one of us make decisions on how much of our income we save (or how much we borrow in order to survive). In fact, this decision is the focus of the next part of the course. The golden rule answers the question: What if we gave the economy the choice of savings rate s - what would they choose? 0 1 2 3 4 5 6 0 0.5 1 1.5 2 ( n + g + δ ) ˜ k ˜ k α s ˜ k α c ss In short, they would choose the value of s , s GR , that maximized steady state consumption. When com- paring steady states, we must keep in mind that higher levels of capital affect both output and depreciation. If the capital stock is below the Golden Rule level ˜ k GR , an increase in the capital stock raises output more than depreciation, so consumption rises. In this case, the production function is steeper than the depreciation line, so the gap between these two curves - which equals steady state consumption - grows as ˜ k increases. By contrast, if the capital stock is above the Golden Rule level, an increase in the capital stock reduces steady state consumption, because the increase in output is smaller than the increase in depreciation. In this case, the production function is flatter than the depreciation line, so the gap between the curves - steady state consumption - shrinks as ˜ k rises. At the Golden Rule level of capital, the production function and the depreciation line have the same slope, and steady state consumption is at its greatest level. 1 No, not that golden rule. Though it is a good rule to live by. 1
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0 1 2 3 4 5 6 0 0.5 1 1.5 2 ( n + g + δ ) ˜ k ˜ k α c GR Solving for the Golden Rule Just like we solved for the steady state in terms of A,δ,s, and α , we can do the same for the golden rule level of capital and savings rate. To do this, notice that ˜ c = (1 - s y And ˜ y = ˜ k α . On the balanced growth path, ˜ k = ( s n + g + δ ) 1 1 - α , so ˜ c = (1 - s ) ± s n + g + δ ² 1 1 - α The golden rule savings rate will maximize steady state consumption, so we can take the derivative of ˜ c with respect to s and set that equal to zero - and the s that satisfies that first order condition will be the golden rule savings rate. ˜ c ∂s = (1 - s ) α 1 - α ± s n + g + δ ² α 1 - α - 1 1 n + g + δ - ± s n + g + δ ² α 1 - α = 0 This implies that: (1 - s ) α 1 - α ± s n + g + δ ² α 1 - α - 1 1 n + g + δ = ± s n + g + δ ² α 1 - α and rearranging, (1 - s ) α 1 - α ± s n + g + δ ² - 1 ± 1 n + g + δ ² = 1 2
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1 - s s = 1 - α α = s = α Also, remember the other way to do this - we can also find the point at which the MPK is equal to the ‘depreciation’ rate: ˜ mpk = α ˜ k α - 1 = ( n + g + δ ) = s = α Some Microeconomics
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This note was uploaded on 02/06/2012 for the course ECON Econ102 taught by Professor D during the Spring '11 term at UCLA.

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ln5-revised - Econ 102 Winter 2012 Lecture Note 5 Solow...

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