Intermediate Macroeconomics, Sciences Po, Answer Key to Problem Set 3

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1 Intermediate Macroeconomics, Sciences Po, 2014 Zsófia Bárány Answer Key to Problem Set 3 1. eoclassical production function: Assume Y = zf (K, ) = zk α 1 α with 0 < α < 1. Does this production satisfy the neoclassical properties? For a production function to satisfy the neoclassical, properties the function must satisfy: (i) constant returns to scale, (ii) positive, but diminishing returns to each factor, and (iii) Inada conditions must hold. To chec for CRS, we need to chec whether zf (λk, λ) = λzf (K, ) for any constant λ > 0. Here zf (λk, λ) = z(λk) α (λ) 1 α = zk α 1 α = λzf (K, ) and there are constant returns to scale. ote: Because of CRS we can write output per worer as a function capital per worer only. Denote capital per worer by = K and output per capita by y = Y, we have y = Y = zf ( K, ) = zf ( K, 1) = zf() = zα, i.e. f() = F ( K, 1) = α. To chec for positive, but diminishing marginal products, we need to investigate the signs of the first and second derivatives of the production function w.r.t. each input (holding all other inputs constant). For the first input, capital: The marginal product of capital is MP K = zf = K αzk α 1 1 α > 0. The second derivate of the production w.r.t. capital is 2 zf = MP K 2 K = α(α K 1)Kα 2 1 α < 0. Similiarly for the other input, labor: MP = zf = (1 α)zkα α, and 2 zf = MP 2 = α(1 α)k α α 1 < 0. So the marginal products of each input are positive, but diminishing. Finally, we need to chec whether the Inada conditions hold, i.e. how the marginal products of a production factor behaves as this factor goes to zero and as it goes to infinity (holding all other inputs constant). Start with capital: lim K MP K = lim K αzk α 1 1 α = lim K αz ( ) 1 α K = 0 and lim K 0 MP K = lim K 0 αz ( ) 1 α K =, so the marginal product of capital goes to zero as capital goes to infinity, and it goes to infinity as capital goes to zero. Liewise for labor: lim MP = lim (1 α)zk α α = lim (1 α)z ( ) K α = 0 and lim 0 MP = lim 0 (1 α)z ( ) K α =, so the marginal product of labor goes to zero as labor 1

2 goes to infinity, and it goes to infinity as labor goes to zero. Hence the Inada conditons hold for capital and labor. These conditions guarantee that production uses an finite non-zero amount of capital and labor. Therefore this function is a neoclassical production function. 2. The Solow Model: Consider the Solow model without technological progress and assume the following production function Y = F (K, ) = K (a) What is the production function in per worer terms? Production function per worer can be derived as y = Y = F (K, ) = F ( ) K, 1 = F (, 1) = = 2 where the third equality maes use of constant returns to scale in the same fashion as in question 1. (b) Show what determines the steady state. Consider first a general production function Y = zf (K, ). Define capital per worer as = K/. ote capital per worer next period is = K /. Start with the capital accumulation equation: K = I+(1 d)k = S+(1 d)k = s Y +(1 d)k = s zf (K, )+(1 d)k Divide by to transform this into per capita terms: K = s z F (K, ) + (1 d) K Expand the left-hand side by mutliplying by, and collect terms as follows: (1 + n) K F (K, ) = s z + (1 d) K (1 + n) = s zf() + (1 d) where the left-hand side uses the fact that K = K = (1 + n). 2

3 In the steady state capital per worer is constant, = =. Therefore (1 + n) = s zf( ) + (1 d) (n + d) }{{} = s zf( ) }{{} brea-even investment actual investment The level of capital per worer stays constant in the steady state, as the amount of actual investment is exactly such that it compensates for depreciation and the growth in population, i.e. for the breaeven investment. Using the functional forms given here, the steady state condition is (n + d) = s ( ) 1 2 which gives = ( s 2. n+d) In the steady state, since capital per worer is constant, also output per worer is constant, y = ( ) 1 2 = s. n+d Aggregate capital K = and aggregate output Y = y grow in steady state at the same rate as the population, i.e. at rate n. (c) Suppose that the economy is in a steady state and that some of its capital stoc is destroyed because of a natural disaster. What are the long-run effects of this on the quantity of capital per worer and on output per worer?. The long-run equilibrium is not changed by an alteration of the initial conditions. If the economy started in a steady state, the economy will return to the same steady state. The formulas from the previous part apply, so that we see that the level of capital per worer in the steady state does not depend on the actual level of capital (but is a function of parameters only). Moreover, when is unchanged, then also y stays the same. (d) What are the short-run effects of this natural disaster? In the short run, following the destruction of part of the capital stoc, does aggregate output grow at a rate higher or lower than the growth rate of the labor force? Initially, the growth rate of the capital stoc will exceed the growth rate of the labor force. The faster growth rate in capital continues 3

4 i (d + n) szf() 1 Figure 1: Transition after Destruction of Capital until the steady state is reached (where aggregate capital will grow at rate n). As capital is destroyed the marginal product of the remaining capital increases, which stimulates further investment and higher growth. Graphically, we move to the left of, say to 1, where the savings curve lies above the depreciation line. On the transition path to the steady state, will grow until it reaches again. Aggregate capital, K = will therefore grow by more than just the population growth rate. Formally, the growth rate of aggregate capital is K K = = 1 = 1 n + K.1 In steady state = =, and the growth rate of aggregate capital is n, but during the transition > 0, which contributes further to the growth in aggregate capital. 3. The Golden Rule: Continue with the assumptions of question 2, but now assume that there are two countries. In both countries there is no population growth and 5% of capital depreciates every year. Assume further that country A saves 10% of output each year and country B saves 20% of output each year. (a) Find the steady state level of capital per worer for each country. Then, find the steady state levels of output per worer and consumption per worer. 1 An alternative way of deriving this is to start with K =, taing the natural logarithm ln(k) = ln() + ln() and the difference over time to get ln(k ) ln(k) = ln( ) ln() + ln( ) ln(), implying K K K + n. 4

5 With n = 0, steady state capital per worer is = (s/d) 2. Substituting s A = 0.1 and s B = 0.2 the steady-state levels of capital per worer are A = 4 and B = 16. Output per worer can be derived from the production function: y = 1 2. Therefore y A = 2 and y B = 4. Finally, the steady-state consumption per worer is given by c = (1 s)y. Therefore c A = 1.8 and c B = 3.2. We have shown that the country with the higher saving rate will reach a higher level of output per worer in the long run (will be richer in the steady state). Furthermore, while country B temporarily sacrifices consumption in order to save a larger fraction of income, she ends up consuming more than country A in the long run. (b) Assume that there is a third country C ruled by a dictator, who decides that almost all of production will be saved and invested for future generations. The saving rate is now s = 90%. What will be the country s long-term consumption level? Will future generations cheerfully build monuments to commemorate him (of course, assuming it won t be run by another dictator who orders its subjects to do so)? Explain two economic mechanisms by means of which the higher savings rate affects consumption. Using the formulae from above we have C = 324, y C c C = 1.8. = 18, and One can observe that such a large saving rate must lie beyond its golden rule level since country C cannot afford to consume as much as country B (which saves a lower fraction of income). There are two forces behind this result that partially offset each other, as can be seen from the formula for the steady-state consumption: c = (1 s)y = (1 s) 1 2 = (1 s) ( (s d ) 2 ) 1 2 = (1 s) s d 5

6 First, higher savings imply higher investment, which translates in higher capital stoc, output and consumption. However, due to diminishing returns to capital, this mechanism gets weaer as capital becomes abundant. The other force is that with higher savings rate, the amount of resources left for consumption is lower. This mechanism persists as we increase the savings rate so that once it gets beyond a certain point (the golden rule level) the cost in terms of consumption of increasing it further outweighs the benefit of an additional unit of invested capital. Additional saving becomes counter-productive. (c) Find the level of savings that maximises consumption in the steady state. Intuitively, the optimal level for the savings rate is such that the additional unit of invested capital adds as much to consumption (through higher production) as will be needed to mae up for the loss of consumption caused by saving a higher fraction of income. Recall that in the steady state all savings covers brea-even investment (depreciation) so that the above stated condition can be rephrased by saying that the savings rate is in its optimum when the marginal product of capital equals the marginal increase in investment (depreciation). ote that this is a stronger condition than we require for any steady state. In terms of Figure 2, we are choosing g (itself being a function of the optimal saving rate s g ) that maximizes consumption c = y sy. Since we are looing for the steady-state level of consumption, we can write c = y d = f( ) d. We have shown in part (b) that steady state capital is a function of the saving rate. One approach would be to substitute the formula for and solve for s g directly but it is instructive first to solve for and then derive s g. The first-order condition is c = f ( ) d = 0 f ( ) = d which confirms the intuitive explanation given above: the level of that maximises consumption can be found at the point where the 6

7 a higher fraction of income. Recall that in the steady state all savings covers brea-even investment (depreciation) so that the above stated condition can be rephrased by saying that the savings rate is in its optimum when the marginal product of capital equals the marginal increase in investment (depreciation). ote that this is a stronger condition than we require for any steady state. In terms of Figure 3, we are choosing g (itself being a function of the optimal saving rate s g ) that maximizes consumption c = y sy. d f() c g s g f() d g g Figure 2: The golden rule capital stoc Since we are looing for the steady-state level of consumption, we can write c = y d = f( ) d. We have shown in part (b) that steady state slope of the production function (LHS) is the same as the slope of the depreciation line (RHS). 5 Using the production function given in this question we have = 2 d which implies that = 1. Using the formula for steady-state 4d 2 capital from part (b): = (s/d) 2 we can find the golden-rule saving rate: = 1 ( sg d 4d 2 ) ( 2 1 = 2d s g d = 1 2d s g = 0.5 ) 2 7