1. 7. THE SOLOW MODEL OF GROWTH
GROWTH ECONOMICS and Fund-raising in international cooperation SECS-P01, CFU 9 Economics for Development academic year
7. THE SOLOW MODEL OF GROWTH
Roberto Pasca di Magliano
Fondazione Roma Sapienza-Cooperazione Internazionale

2. Introduction
The technical progress in the Solow model represents the exogenous source of the growth as well the exit-way to the steady state situation designed by the Classical theories.
Before explaining the reasons moving the technical progress we have to deal with the its measurement.
The 'approach' growth accounting 'is proposed to decompose the growth rate of the economy in the contributions of both 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
The model has 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 has been built on the basis of the Cobb-Douglas (Y=f(K,L) production model by adding a theory of capital accumulation
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 can be described as follows:
The increase of production is due to:
increase in the factors of the production, capital (K) and labor (L) (endogenous variables).
Increase of the technological change (exogenous variable) -> shape of the production function changes over time (t).

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

6. The representation of the output through isoquants
The production increases for the following reasons:
increase of only one factor. In this case, given the diminishing returns of the single factor, the production increase will stop soon or after.
Increase of both production factors by considering the hypothesis of constant, increasing or decreasing returns to scale .
The Solow model assumes constant returns to scale by using the theory of marginal distribution

7. Building the Solow Model: fair 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: fair market supply
The slope of this function is the marginal product of capital per worker. MPK = f(k+1)–f(k) k y
y=f(k)
It tells us the change in output per worker that results when we increase the capital per worker by one.
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. Then: y = (1–s)y + I …rearranging, I = s*y …so investment per worker equals savings per worker

10. Steady State Equilibrium
In order to reach the steady state equilibrium, he operates as it follows :
substituting f(k) for (y), the investment per worker function (i = s*y) becomes a function of capital per worker (i = s*f(k)).
then, 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

11. At this point, dKt = sYt, so
The Solow equilibrium diagram production function, capital accumulation (Kt on the x-axis)
Investment, Depreciation
Capital, Kt
At this point,
dKt = sYt, so

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

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

14. Conclusion
The Solow Growth model is a dynamic model that shows:
how endogenous variables capital per worker and output per worker are affected by the exogenous variable savings;
how capital depreciation enters in the model,
the effects that initial capital allocations have on the time paths toward equilibrium.