Understanding Economic Growth Rates Away from Steady State in the Solow Model, Study notes of Economics

Download Understanding Economic Growth Rates Away from Steady State in the Solow Model and more Study notes Economics in PDF only on Docsity! Fall Semester ’05-’06 Akila Weerapana Lecture 7: Growth Away From Steady State I. OVERVIEW • In the last lecture we examined some properties of the steady state. In the lecture before that, we drew the graphs for the time paths of k, y, K and Y in various comparative statics exercises. • One of the issues left unresolved was the shape of the graph during the transition period. Essentially, we need to have a better explanation of the growth rates of these variables away from steady state. Today’s lecture takes a closer look at the behavior of economies away from steady state. • This is important because many of the interesting growth stories in the world are countries in transition. Consider China whose economy grows at 10% a year. China couldn’t be in steady state because at a 10% growth rate their income would double every 7 years and be 16,384 times larger than today in a hundred years time. So to understand the behavior of economic growth in China, we need to dig deeper into the Solow model. II. GROWTH RATES AWAY FROM STEADY STATE • Let’s take another look at what the Solow model has to say about growth rates of y and Y away from steady state. In order to do this we look at a modified Solow diagram obtained from the capital accumulation equation. k̇ = sy − (n + δ)k ⇒ k̇ k = sy k − (n + δ) k̇ k = skα k − (n + δ) k̇ k = s k1−α − (n + δ) • The graph for s k1−α is downward sloping since an increase in k raises the value of the denom- inator. To put it in terms of derivatives, we can show that d(sk α−1) dk = s(α− 1)k α−2 < 0 • Furthermore, we can calculate the 2nd derivative as positive because d 2(skα−1) dk2 = s(α−1)(α− 2)kα−3 > 0. In other words, the shape of this curve is convex. • You can also show that limk→0 sk1−α =∞ and limk→∞ s k1−α = 0 • (n + δ) is of course just a horizontal line that doesn’t change with k̃ • The picture below graphs these two lines. The difference between the s k1−α and the (n + δ) lines (indicated by the dotted lines in the figure below) shows the growth rate of k, i.e. gives us the value of k̇k - 6 kt (n + δ) s k1−α k∗ qqqqqq qqqqq qqq k̇k > 0 k̇ k < 0 ---  • In this diagram, for any level of capital per worker (k), the gap between the downward sloping curve and the horizontal line is the growth rate of capital per worker k̇k . • We can conclude the following: 1. At the steady state, the point at which the two lines cross, k̇k = 0 2. As the economy approaches steady state from below, k̇k > 0 but getting closer to 0. 3. As the economy approaches steady state from above, k̇k < 0 but getting closer to 0 • This is what gives us the concave and convex shapes in drawing the time paths of variables. When moving from a lower (constant) steady state to a higher (constant) steady state, we should see a positive growth rate that is gradually approaching zero, i.e. a concave shape. • Conversely when moving from a higher (constant) steady state to a lower (constant) steady state, we should see a negative growth rate that is gradually approaching zero, i.e. a convex shape. • Using the fact that y = kα we can also derive the following results for the growth rate of y 1. At the steady state, the point at which the two lines cross, ẏy = 0 2. As the economy approaches steady state from below, ẏy > 0 but getting closer to 0. 3. As the economy approaches steady state from above, ẏy < 0 but getting closer to 0 • Basically, the Solow model predicts that the further below steady state an economy is, the faster output per worker will grow and the further above steady state an economy is, the slower output per worker will grow. • Finally, using the fact that ẎY = ẏ y +n we can also derive the following prediction: the further below steady state an economy is, the faster total output will grow and the further above steady state an economy is, the slower total output will grow.