Solutions to Exercises in Introduction to Economic Growth
Charles I. Jones (with Chao Wei and Jesse Czelusta) Department of Economics U.C. Berkeley Berkeley, CA 94720-3880
September 18, 2001
2 The Solow Model
Exercise 1. A decrease in the investment rate. A decrease in the investment rate causes the s˜ curve to shift down: at any y ˜the investment-technology ratio is lower at the new rate of savgiven level of k, ing/investment. Assuming the economy began in steady state, the capital-technology ratio is now higher than is consistent with the reduced saving rate, so it declines gradually, as shown in Figure 1. Figure 1: A Decrease in the Investment Rate
~ s’y ~ s’’ y
The log of output perworker y evolves as in Figure 2, and the dynamics of ˙ ˜ ˜ ˜ the growth rate are shown in Figure 3. Recall that log y = α log k and k/k = ˜ ˜ s k α−1 − (n + g + d). The policy permanently reduces the level of output per worker, but the growth rate per worker is only temporarily reduced and will return to g in the long run.
Figure 2: y(t)
Figure 3: Growth Rate of Output perWorker
. y/ y
3 Exercise 2. An increase in the labor force. The key to this question is to recognize that the initial effect of a sudden increase in the labor force is to reduce the capital-labor ratio since k ≡ K/L and K is ﬁxed at a moment in time. Assuming the economy was in steady state prior to the increase in labor force, k falls from k ∗ to some new level k1 . Noticethat this is a movement along the sy and (n + d)k curves rather than a shift of either schedule: both curves are plotted as functions of k, so that a change in k is a movement along the curves. (For this reason, it is somewhat tricky to put this question ﬁrst!) ˙ At k1 , sy > (n + d)k1 , so that k > 0, and the economy evolves according to the usual Solow dynamics, as shown in Figure 4. Figure 4:An Increase in the Labor Force
(n + d) k sy
In the short run, per capita output and capital drop in response to a inlarge ﬂow of workers. Then these two variables start to grow (at a decreasing rate), until in the long run per capita capital returns to the original level, k ∗ . In the long run, nothing has changed! Exercise 3. An income tax. Assume that the government throwsaway the resources it receives in taxes. Then an income tax reduces the total amount available for investing and shifts the investment curve down as shown in Figure 5.
Figure 5: An Income Tax
~ (n+g+d) k ~ sy ~ s y(1- τ)
~ ** k
The tax policy permanently reduces the level of output per worker, but the growth rate per worker is only temporarily lowered. Notice thatthis experiment has basically the same results as that in Exercise 2. For further thought: what happens if instead of throwing away the resources it collects the government uses all of its tax revenue to undertake investment? Exercise 4. Manna falls faster. Figure 6 shows the Solow diagram for this question. It turns out, however, that it’s easier to answer this question using the transitiondynamics version of the ˙ ˜ ˜ diagram, as shown in Figure 7. When g rises to g , k/k turns negative, as shown in ˙ Figure 7 and A/A = g , the new steady-state growth rate. To see what this implies about the growth rate of y, recall that ˙ ˜ ˙ ˙ k y ˙ y A ˜ = + =α +g. ˜ y y A ˜ k So to determine what happens to the growth rate of y at the moment of the change ˙ ˜ ˜ in g, we have to determine what happensto k/k at that moment. As can be seen in Figure 7, or by algebra, this growth rate falls to g − g < 0 — it is the negative of the difference between the two horizontal lines. Substituting into the equation above, we see that y/y immediately after the ˙
Figure 6: An Increase in g
~ (n+g’+d)k ~ (n+g+d)k
Figure 7: An Increase in g: Transition Dynamics...
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