The Solow Model. Econ 4960: Economic Growth

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1 The Solow Model All theory depends on assumptions which are not quite true That is what makes it theory The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive Solow (1956, Introduction) Nothing is less real than realism Details are confusing It is only by selection, by elimination, by emphasis, that we get at the real meaning of things Georgia O Keeffe 1

2 1 Production Technology Production function is Cobb-Douglas: Y = F K, L = K a L -a ( ) 1 Firms are competitive and maximize profit taking prices as given: Max (F(K,L) rk wl) First order conditions: F Y w = = (1-a ) L L F Y r = = a K K 1 Production Technology (cont d) Two Properties: 1 wl + rk = Y Zero profit! 2 Labor share: Capital share: wl / Y = 1- Transform production function into per-capita terms: Define: y=y/l and k=k/l We have : y= k a rk / Y a = a 2

3 Cobb-Douglas Production: Inada Conditions Capital is essential: Diminishing marginal product: lim f '(x) < δ x High marginal product at zero lim f '(x) > δ x 0 y = f (0) = 0 3

4 Inada Conditions (cont d) Cobb-Douglas satisfies the Inada conditions, which rules out several types of behavior Some of the ruled-out cases are strange, others are interesting We will discuss some of these cases later Violations of Inada 4

5 Capital and Labor Share over Time 2 Capital accumulation Solow makes the behavioral assumption that Consumers save a constant fraction of their income Therefore: where K = sy dk dk K = dt Finally, the economy is closed to international trade 5

6 Capital accumulation per person Simple trick: k º K/ L log k = log K -log L k K L L = - further assume: = n k K L L We have: K Y K = sy -dk Þ = s -d K K k Y Þ = s -n - d = sy k - n + d k K ( ) Þ k = sy - n + d k ( ) / ( ) Steady State Steady state is a key concept in economics A steady state is a point such that if the system is at that point, it remains there forever Solve for by setting k = 0 k * a ( ) ( ) k = sy- n+ d k = 0 Þsk - n+ d k = 0 ( ) ( ) ( ) 1/ 1 -a a/ 1-a * æ s ö * * a æ s ö Þ k = ç and y = k = ç èn+ d ø èn+ d ø k * 6

7 (The famous!) Solow Diagram Comparative Statics From the steady state equations: A rise in investment rate increases y* A rise in population growth rate reduces y* These are consistent with empirical evidence (figs 26, 27) A rise in depreciation rate reduces y* y * æ s ö = ç èn+ d ø ( -a) a/1 7

8 Transitional Dynamics A striking implication of Solow s model is that that there is no growth in the long-run! This is what a steady state means after all There is only growth temporarily, until you converge to y* To see transitional dynamics, note: k k ( ) = - + a -1 sk n d Transitional Dynamics 8

9 Transition after an increase in savings rate Transitional Dynamics in Discrete Time 9

10 Absolute Convergence or the Lack Thereof Absolute versus Conditional Convergence However, the assumption that all countries have the same technology, and tastes is very strong If one instead focuses on countries with similar characteristics, such as OECD economies you get a different picture Barro and Sala-i Martin actually estimated that when one controls for differences in characteristics, countries (with same human capital and life expectancy) converge to each other by about 2 percent per year Aside: If you want to see the most colorful growth economist, check out Sala-i Martin s website: 10

11 Conditional Convergence: Seems more plausible Discussion questions (non-trivial) Suppose that country A has a higher y* than country B Can you tell which one of these countries is likely to grow faster in the short-run (for example, next year)? [Think about different sources of differences in y* Are there any conditions under which you can say something?] What effect does globalization have on convergence in the Solow model? What does the basic Solow Model say about Aid for Africa? What modification do you need to make to alter this conclusion? Suppose we relax Inada condition 1: F(0,L)>0 How does that affect the steady states? 11

12 Taking Stock The standard Solow model: predicts that two countries will have different y* if they differ in s, n, or d That there is no growth in output (and capital) per person in the long-run Only growth happens during transitions to the steady state Growth rate slows down as countries become more developed 12