ECON 1910 Spring 2012 Lind / Willumsen
Short solution proposal to the compulsory assignment in ECON1910 Problem 1: Harrod-Domar vs. Solow. In the Harrod-Domar model a change in the savings rate (s) has a permanent eﬀect on the growth rate of GDP per capita, while in the Solow model a change in the savings rate has only a temporary eﬀect on the growth rate of GDP per capita. Why is this the case? Answer: The main diﬀerence between the Harrod-Domar (HD) model and the Solow model is that HD assumes constant marginal returns to capital, while Solow assumes decreasing marginal returns to capital. The reason that a change in the savings rate has a permanent eﬀect in HD, while only a temporary eﬀect in Solow, is exactly due to the diﬀerences in assumptions on the marginal returns to capital. To see why, assume that we initially are in the steady state in the Solow model, where investments exactly are equal to break-even investments (i.e. the amount of investments syt are equal to (n + δ)kt , the amount of investment that needs to be undertaken in order for the capital stock per capita next period to be the same size as today). If we increase the savings rate in the Solow model from s to s , we will in the next period have more capital per capita than before, as depreciation (δ), population growth (n) and capital today (kt ) are the same, i.e. break-even investments today do not change. This additional capital will generate more output next period (a fraction s of which is saved), but we will also need to invest more next period if we were to keep capital constant at this new level since the new break-even investment level (n + δ)kt+1 > (n + δ)kt since kt+1 > kt . It will now (in period t + 1) also be the case that investments are higher than the new break-even investment level, but less so than last period because the marginal product of capital is lower at the new and higher level of k. As the marginal product of capital decreases as k gets larger, while the ‘cost’ of higher k in terms of higher break-even investments increases linearly with k, the temporary eﬀect on growth of the change in s will gradually level oﬀ until we reach the new steady state, where growth again is 0. Note that the last argument does not hold for the HD model. In the HD model the marginal product of capital per capita is constant, and hence a permanent change in s will have a permanent eﬀect on the growth rate of the economy. Problem 2: Population growth rates in the Solow model. Deﬁne and explain “steady state” in the Solow model. Assume that the economy initially is in the steady state. Analyze the short-run and long-run eﬀects of a change in the population growth rate (n) on per capita GDP growth rates and levels in the Solow model, everything else equal. Answer: Steady state: The state the economy eventually converges to for arbitrary initial conditions, identiﬁed by constant growth rates (here 0). The easiest way to answer this question is to do a graphical analysis of the Solow model, see Figure 1 and the slides accompanying Lecture 4 (Note that on the exam you will have to deﬁne all the variables you are using and explain what the diﬀerent curves in the ﬁgure represent. We do not do that here, however, because all the deﬁnitions and the explanation of the ﬁgure is in the slides accompanying Lecture 1
(n + δ)kt f (kt )
(n + δ)kt sf (kt )
Figure 1: The eﬀect of a change in population growth from n to n in the Solow model 4). In Figure 1 we assume that we initially are in the steady state (k ∗ , y ∗ ). Then the population growth rate changes from n to n . We see that this changes the slope of the line denoting break-even investments—it becomes steeper. The new intersection between the break-even investment curve and the savings curve (the steady state) is now at the point (k ∗ , y ∗ ), and we see that k ∗ < k ∗ and y ∗ < y ∗ . The long term eﬀect of a change in population growth from n to n , where n >...
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