1. POLITICAL ECONOMY OF GROWTH SECS-P01, CFU 9 Finance and Development academic year 2016-17
7. THE SOLOW MODEL
Roberto Pasca di Magliano
Fondazione Roma Sapienza-Cooperazione Internazionale

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

3. Building the 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)

4. Building the Model: goods 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

5. Building the Model: goods market demand
Begining 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 generated: 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.

6. 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).

7. Steady State Equilibrium
In order to reach the stady state equilibrium:
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

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

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

10. Suppose the economy starts at K0:
Investment, Depreciation
Capital, Kt
The red line is above the green at K0:
Saving = investment is greater than depreciation at K0
So ∆Kt > 0 because
Since ∆Kt > 0, Kt increases from K0 to K1 > K0 K0 K1

11. Now imagine if we start at a K0 here:
Investment, Depreciation
Capital, Kt
At K0, the green line is above the red line
Saving = investment is now less than depreciation
So ∆Kt < 0 because
Then since ∆Kt < 0, Kt decreases from K0 to K1 < K0 K1 K0

12. We call this the process of transition dynamics: Transitioning from any Kt toward the economy’s steady-state K*, where ∆Kt = 0 and growth ceases
Investment, Depreciation
Capital, Kt K*
No matter where we start, we’ll transition to K*!
At this value of K, dKt = sYt, so

13. 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?
This would increase the slope of our investment function and cause the function to shift up. k*
s*f(k*)=dk*
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.

14. We can see what happens to output, Y, and thus to growth if we rescale the vertical axis:
Investment, Depreciation, Income
Saving = investment and depreciation now appear here
Now output can be graphed in the space above in the graph
We still have transition dynamics toward K*
So we also have dynamics toward a steady-state level of income, Y* K* Y*
Capital, Kt

15. 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

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 other section the dynamic model is improved in order to include changes in other exogenous variables; population and technological growth.