Roberto Pasca di Magliano Fondazione Roma Sapienza-Cooperazione Internazionale

1. GROWTH ECONOMICSand Fund-raising in international cooperationSECS-P01, CFU 9Economics for Developmentacademic year 2017-18 7. THE SOLOW MODEL OF GROWTH AND TECHNOLOGICAL PROGRESS Roberto Pasca di Magliano Fondazione Roma Sapienza-Cooperazione Internazionale roberto.pasca@uniroma1.it

2. Introduction • The technical progress in the Solow model shows the exogenous cause of the growth of per capita income steady state. • Before explaining the reasons of technical progress we have to deal with the problem of its measurement. • The “growth accounting approach” is proposed to decompose the growth rate of the economy between the contribution of the accumulation of factors of production and technical progress.

3. Model Background The Solow growth model is the starting point to determine why growth differs across similar countries it has build on the basis of the Cobb-Douglas production model by adding a theory of capital accumulation It was developed in the mid-1950s by Robert Solow of the MIT and it was the basis for the Nobel Prize he received in 1987 It is based on the hypothesis that the accumulation of capital is a possible engine of long-run economic growth

4. The Production Function The product of an economy isrepresented by: The increase of production may be due to: • increase in the factors of production, capital (K) and labor (L). • technologicalchange. The shape of the production functionchanges over time (t).

5. Representation of the output through isoquants The isoquants provide a geometric representation of the production function: the pairs of K and L produce a given output Y.

6. The representation ofthe output through isoquants The production increases for the following reasons: • Increase of one of the factors. Given the diminishing returns of the single factor, this increase will stop. • Increase of both factors. • Constant returns, increasing or decreasing. The Solow model assumes constant returns to scale by using the theory of marginal distribution

7. Building the Solow Model: goods market supply We begin with a production function and assume constant returns: Y=F(K,L) so… zY= f (zK, zL) By setting z =1/L it is possible to create a per worker function. Y/L=f (K/L,1) So, output per worker is a function of capital per worker: y=f(k)

8. Building the Model: goods market supply y y=f(k) k • The slope of this function is the marginal product of capital per worker.MPK = f(k+1)–f(k) • It tells us the change in output per worker that follows an increase of one point of per worker capital Change in y Change in k

9. Building the Model:goods market demand Beginning with per worker consumption and investment (Government purchases and net exports are not included in the Solow model), the following per worker national income accounting identity can be obtained: y = c + I Given a savings rate (s) and a consumption rate (1–s) a consumption function can be generated as follows: c = (1–s)y …which is the identity. Theny = (1–s)y + I …rearranging,I = s*y …so investment per worker equals savings per worker

10. Steady State Equilibrium The Solow model long run equilibrium occurs at the point where both (y) and (k) are constant The endogenous variables in the model are y and k The exogenous variable is (s).

11. Steady State Equilibrium In order to reach the steady state equilibrium, we operate as it follows : By substituting f(k) for (y), the investment per worker function (i = s*y) becomes a function of capital per worker (i = s*f(k)). By adding a depreciation rate (d) The impact of investment and depreciation on capital can be developed to evaluate the need of capital change: dk = i – dk …substituting for (i) dk = s*f(k) – dk

12. Investment, Depreciation At this point, dKt = sYt, so Capital, Kt The Solow Diagram equilibriumproduction function, capital accumulation (Kt on the x-axis)

13. Depreciation: d K Investment: s Y Investment, depreciation Net investment K0 K* Capital, K The Solow DiagramWhen investment is greater than depreciation, the capital stock increase until investment equals depreciation. At this steady state point, dK = 0

14. Changing the exogenous variable: savings Investment, Depreciation dk s*f(k) s*f(k*)=dk* k k* • We know that steady state is at the point where s*f(k)=dk s*f(k*)=dk* s*f(k) • What happens if we increase savings? • This would increase the slope of our investment function and cause the function to shift up. k** • This would lead to a higher steady state level of capital. • Similarly a lower savings rate leads to a lower steady state level of capital.

15. Investment, depreciation, and output Y0 Y* Investment: s Y Depreciation: d K Consumption K0 K* Capital, K The Solow Diagram with OutputAt any point, Consumption is the difference between Output and Investment: C = Y – I Output: Y

16. Conclusion The Solow Growth model is a dynamic model that allows us to see how our endogenous variables capital per worker and output per worker are affected by the exogenous variable savings. We also see how parameters such as depreciation enter the model, and finally the effects that initial capital allocations have on the time paths toward equilibrium. In another section the dynamic model is improved in order to include changes in other exogenous variables; population and technological growth.