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# Solow_qa - Solow Growth Sample Questions David Lagakos May...

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Solow Growth Sample Questions David Lagakos, May 5, 2007 Question 1 - Setting up the Solow Model (a) Let the production function be Cobb-Douglas with capital share α and productiv- ity term A , where Y t is output, K t is capital, and L t is labor at time t . Let the savings rate be s , the population growth rate be n , and the depreciation rate be δ . Let con- sumption and investment be C t and I t . Write down the production function, the law of motion for labor, the law of motion for capital, the resource constraint, and the investment equation that describe the Solow Growth Model. Describe in plain English what each equation means. Ans. 1. Production Function Y t = AK α t L 1 - α t ”Output is produced using physical capital and labor inputs, using a Cobb-Douglas production function.” 2. Law of Motion for Labor L t +1 = L t (1 + n ) ”The population of this economy grows each period at a rate n . Each member of the population is assumed to provide labor for production.” 3. Law of Motion for Capital K t +1 = K t (1 - δ ) + I t 1

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”The capital available for production next period equals depreciated capital this pe- riod (first term) plus today’s investment (second term).” 4. Resource Constraint I t + C t = Y t ”All output produced this period is either consumed now or invested for the future.” 5. Investment Equation I t = sY t ”In this economy we assume that a fixed fraction s of output is saved each period and invested in physical capital for the next period.” (b) Derive the per-capita resource constraint, the per-capita law of motion for capital, and the per-capita investment equation. Express per-capita variables in lower-case variables. Ans. a. Per-capita production Function y t = Ak α t b. Per-capita law of motion for Capital Starting from the law of motion for capital, we divide both sides by L t to get K t +1 L t = k t (1 - δ ) + i t . 2
Using the law of motion for labor we know that L t = L t +1 / (1 + n ) , which implies that K t +1 L t +1 (1 + n ) = k t (1 - δ ) + i t or more simply k t +1 (1 + n ) = k t (1 - δ ) + i t . 4. Per-capita Resource Constraint i t + c t = y t 5. Per-capita Investment Equation i t = sy t (c) Derive the ’required investment’ curve using the per-capita law of motion in steady state. Provide an economic interpretation. Ans. In steady state k t +1 = k t . Let’s drop the time subscripts then to get k (1 + n ) = k (1 - δ ) + i. Canceling a k from both sides and re-arranging gives i = k ( δ + n ) . The interpretation is that in order to keep capital per worker constant, investment 3

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has to exactly offset population growth plus the depreciation of existing capital. This investment level is the required amount of investment to maintain a steady state level of capital per worker. (d) Plot this equation with k on the x -axis and i on the y -axis. What’s the slope of the line? Describe what happens to k for points on this graph (i.e. i , k pairs) above and below the line.
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Solow_qa - Solow Growth Sample Questions David Lagakos May...

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