The Solow Model. DeÞnition 2: A balanced growth path is a situation where each variable in the model is growing at a constant rate.

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1 DeÞnition 1: The steady state level of capital per unit of effective labour, k, is the level of capital per unit of effective labour that equates break even investment and actual investment i.e., sf(k )=(n+g+δ)k. DeÞnition 2: A balanced growth path is a situation where each variable in the model is growing at a constant rate. DeÞnition 3. The golden rule level of capital per unit of effective labour, k G, is the steady state level of capital per unit of effective labour that would maximize the steady state value of consumption per unit of effective labour. 1

2 Speed of convergence here we will examine how to determine the rate at which k approaches k remember k(t) is a function of k(t) since k(t) =sf(k(t)) (n + g + δ)k(t) so k = k(k) and when k = k, k =0 to determine the behaviour of k we will examine a Þrst order Taylor series approx. of k(k) around the point k = k (intuitively this 1 st order approx. is approximately the function around k = k by a line. 2

3 The 1 st order Taylor series approx. h(y) around the point x is h(y) ' h(x)+h 0 (x)(y x) of a function then k(k) ' k(k )+ k(k) k k=k = 0+ k(k ) k (k k ) (k k ) Assume a Cobb-Douglas production function, that is f(k) =k α, then k(k ) k = sf 0 (k ) (n + g + δ) =sαk α 1 (n + g + δ) Since at k s = (n+g+δ)k, then (n + g + δ) k α k(k ) k = (n+g+δ) k α 1 αk α 1 3

4 so k(k ) k =(α 1)(n + g + δ) subbing into the original equation we get: k(k(t)) ' (α 1)(n + g + δ)(k(t) k ) = λ(k(t) k ) where λ =(1 α)(n + g + δ) in the vicinity of the balanced growth path, the growth rate of (k(t) k ) is approximately constant and equal to λ. toseethisnotethatthegrowthrateof(k(t) k ) k(t) is (k(t) k and we know from above this is approx ) equal to λ it follows that (k(t) k ) ' e λt (k(0) k ) 4

5 the model also predicts that (y(t) y ) ' e λt (y(0) y ) to see this formally remember y(t) = f(k(t)) so y(t) =f 0 (k(t)) k(t) then expanding around k = k we get y(t) ' f(k )+f 0 (k )(k(t) k ) (y(t) y ) ' f 0 (k )(k(t) k ) so (y(t) y ) ' e λt (y(0) y ) 5

6 Example: Population growth rate-2% Growth in output per worker-2% Depreciation rate-2% α =1/3 then λ =(1 α)(n + g + δ) =(2/3)(6%) = 4% k and y move 4% of the remaining distance towards k and y each year Howlongdoesittaketomove1/2 way to the balanced growth path values e tλ = 0.5 ' tλ = ln(0.5) t = ln(0.5) 0.04 y(t) y y(0) y ' ' k(t) k k(0) k so 6

7 EVALUATION OF THE SOLOW MODEL. Qualitative implications of the Solow model: 1. Per capita income levels are higher in economies with higher investment (savings) rates. 2. Per capita income levels are lower in economies with higher population growth rates. 3. Long run growth rates of per capita variables are not affected by "policy" changes (changes in savings rates, immigration etc.). 4. In the long run, changes in parameters have only "level" effects. 7

8 5. Economies with same saving rates, technology and demographic parameters but different capital stocks will eventually converge to the same per capita income level. 6. On the balanced growth path, income per capita grows at a constant rate. 7. On the balanced growth path, the return of capital is constant and the wage rate grows at a constant rate (g). 8. On the balanced growth path, capital-output ratio is constant. 9. On the balanced growth path, the shares of capital and labour are constant. 8

9 Quantitative implications of the Solow model: Can the Solow model explain the "magnitude" of the difference in income per capita observed across countries? Two sources of variation in income per capita: Differences in capital per worker across countries. Example: y=k α and α =1/3 If y Rich /y Poor =10 ³ k Rich k =10 k Rich P oor 1/3 1000!!! k P oor = This difference in capital stocks would imply a huge difference in returns of capital: r Rich r P oor = ³ k Rich k P oor 2/3 = /3 = 1 100, that is r Poor =100 r Rich. 9

10 Differences in capital stock across countries can account for a small part of the differences in income per capita observed across countries. Differences in the effectiveness of productivity of labour, A. W e can interpret A as education and skills of labour force, the strength of property rights, the quality of infrastructure, or a combination of all of these forces. 10