1. Chapter 7 Appendix: The Solow Growth Model
© 2015 Pearson Education, Ltd.

2. Appendix Outline 20A.1 The Three Building Blocks of the Solow Model
7A The Solow Growth Model
Appendix Outline
20A.1 The Three Building Blocks of the Solow Model
20A.2 Steady-State Equilibrium in the Solow Model
20A.3 Determinants of GDP
20A.4 Dynamic Equilibrium in the Solow Model
20A.5 Sources of Growth in the Solow Model
20A.6 Calculating Average (Compound) Growth Rates
© 2015 Pearson Education, Ltd.

3. There are three building blocks of the Solow growth model.
7A The Solow Growth Model
Key Ideas
There are three building blocks of the Solow growth model.
The Solow growth model can be solved for a steady-state equilibrium.
In the Solow growth model, increases in the saving rate, human capital, and technology increase the level of real GDP.
© 2015 Pearson Education, Ltd.

4. In the Solow growth model, the steady-state equilibrium is dynamic.
7A The Solow Growth Model
Key Ideas
In the Solow growth model, the steady-state equilibrium is dynamic.
In the Solow growth model, sustained economic growth can be achieved only with increases in technology.
The compound growth formula is used to calculate average annual growth rates.
© 2015 Pearson Education, Ltd.

5. 7A.1 The Three Building Blocks of the Solow Model
The aggregate production function—the first block of the Solow model—determines the level of real GDP:
Instructor: Y is GDP, K is the physical capital stock, H is the total efficiency units of labor, F() is a mathematical function, and A is an index of technology.
© 2015 Pearson Education, Ltd.

6. An equation for physical capital accumulation:
7A.1 The Three Building Blocks of the Solow Model
An equation for physical capital accumulation:
where K is the stock of capital, and I is the flow of new investment.
Instructor: You may want to talk about the distinction between stocks and flows.
© 2015 Pearson Education, Ltd.

7. 7A.1 The Three Building Blocks of the Solow Model
Assuming a constant depreciation rate, the physical capital accumulation becomes:
Instructor: The first term, (1–d) x K(last year), is the capital stock that didn’t depreciate from last year. The second term, I, is the new flow of capital.
© 2015 Pearson Education, Ltd.

8. where S is the constant saving rate.
7A.1 The Three Building Blocks of the Solow Model
Saving by households:
where S is the constant saving rate.
© 2015 Pearson Education, Ltd.

9. Total output Y is divided between C and I:
7A.1 The Three Building Blocks of the Solow Model
Total output Y is divided between C and I:
Instructor: The red curve shows the aggregate production function. It is concave in shape due to diminishing returns in physical capital. The green curve shows the relationship between investment and the capital stock. It too is concave in shape since investment is simply a fixed percentage of income.
The distance up to the investment curve measures saving by households, S. The distance between the investment curve and the production function measures consumption by households, C, which is equal to Y – S.
Exhibit 7A.1 Aggregate Income and Aggregate Saving
© 2015 Pearson Education, Ltd.

10. Steady-state equilibrium
7A.2 Steady-State Equilibrium in the Solow Model
Steady-state equilibrium
An economic equilibrium in which the physical capital stock remains constant over time:
Instructor: This equation is simply the definition of a steady state. The next slide is a derivation of the steady-state condition.
© 2015 Pearson Education, Ltd.

11. 7A.2 Steady-State Equilibrium in the Solow Model
A steady-state equilibrium occurs when new investment is equal to depreciation:
I = depreciation
s x Y = d x K
s x A x F (K,L) = d × K
Instructor: This equation is a derivation of the steady-state condition. Tell the students that given a function form for the production like a Cobb-Douglas function, one could solve for this a steady-state capital stock, K*.
© 2015 Pearson Education, Ltd.

12. 7A.2 Steady-State Equilibrium in the Solow Model
Exhibit 7A.2 Steady-State Equilibrium in the Solow Model
© 2015 Pearson Education, Ltd.

13. 7A.3 Determinants of GDP
An increase in either the saving rate, s, or the stock of human capital, H, will increase the steady-state level of GDP.
© 2015 Pearson Education, Ltd.

14. 7A.3 Determinants of GDP
Instructor: The economy begins at an original steady state at K*, with s × Y* investment. An increase in the saving rate from s to s’ rotates the investment line up. As a result, the economy moves to a new steady state at K**, with s’ × Y** investment.
Exhibit 7A.3 The Impact of the Saving Rate on the Steady-State Equilibrium
© 2015 Pearson Education, Ltd.

15. 7A.3 Determinants of GDP
Instructor: The economy begins at an original steady state at K*, with s × Y* investment. An increase in human capital from H0 to H1 rotates the investment line up. As a result, the economy moves to a new steady state at K**, with s × Y** investment.
Exhibit 7A.4 Change in the Steady-State Equilibrium Resulting from an Increase in the Human Capital of Workers
© 2015 Pearson Education, Ltd.

16. A dynamic equilibrium traces the behavior of the economy over time.
7A.4 Dynamic Equilibrium in the Solow Model
A dynamic equilibrium traces the behavior of the economy over time.
Suppose the economy begins with a physical capital stock of K0 < K*.
Instructor: A steady-state equilibrium is a single point where investment = depreciation. A dynamic equilibrium is a path (of physical capital stock and GDP levels) that will be realized over time.
© 2015 Pearson Education, Ltd.

17. 7A.4 Dynamic Equilibrium in the Solow Model
Instructor: At K0 < K*, investment on the green line exceeds depreciation on the red line. As a result, there is positive capital accumulation until the investment rate falls to the depreciation rate at the new steady state.
Exhibit 7A.5 Dynamic Equilibrium in the Solow Model
© 2015 Pearson Education, Ltd.

18. Remember that we are looking for sources of sustained growth.
7A.5 Sources of Growth in the Solow Model
Increases in the saving rate, s, is not a source of sustained growth in real GDP.
Why? Increases in saving shift the investment curve up and thus provide an increase in the level of GDP.
Remember that we are looking for sources of sustained growth.
© 2015 Pearson Education, Ltd.

19. 7A.5 Sources of Growth in the Solow Model
Instructor: An increase in the saving rate from s to s’ to sMAX rotates up the investment curve but only generates an increase in the level of GDP.
Exhibit 7A.6 Three Economies with Different Saving Rates in the Solow Model
© 2015 Pearson Education, Ltd.

20. Technological progress is a source of sustained growth in real GDP.
7A.5 Sources of Growth in the Solow Model
Technological progress is a source of sustained growth in real GDP.
Why? An increase in technology, A, raises productivity, thus allowing physical and human capital to produce more output.
As a result, technology progress (or constant growth in technology) will lead to sustained increases or growth in real GDP.
© 2015 Pearson Education, Ltd.

21. 7A.5 Sources of Growth in the Solow Model
Instructor: As technology improves, the aggregate production function shifts up, and equilibrium physical capital stock and GDP increase gradually.
Exhibit 7A.7 Sustained Growth Driven by Technological Change
© 2015 Pearson Education, Ltd.

22. At steady-state, investment = depreciation:
7A.5 Sources of Growth in the Solow Model
Another prediction of the Solow growth model is that the ratio of the physical capital stock to GDP should be constant through time.
At steady-state, investment = depreciation:
© 2015 Pearson Education, Ltd.

23. 7A.5 Sources of Growth in the Solow Model
Instructor: The exhibit plots the historical evolution of the value of the physical capital stock to GDP in the U.S. economy. The ratio of the physical capital stock to GDP has been at 2 for the past 50 years.
Exhibit 7A.8 The Ratio of Physical Capital Stock to GDP in the United States
© 2015 Pearson Education, Ltd.

24. Catch-up growth leads to increases in the level of real GDP.
7A.5 Sources of Growth in the Solow Model
Catch-up growth is driven by the accumulation of physical and human capital.
Catch-up growth leads to increases in the level of real GDP.
Although it can dramatically raise the level of GDP, catch-up growth is not a source of sustained growth in real GDP.
Instructor: Remind the students that catch-up growth is a path (of physical capital stock and GDP levels) toward a steady state. In other words, catch-up growth is a dynamic equilibrium.
© 2015 Pearson Education, Ltd.

25. Compound growth is the phenomenon whereby growth builds on growth.
7A.6 Calculating Average (Compound) Growth Rates
Compound growth is the phenomenon whereby growth builds on growth.
Alternatively, compound growth is the earning of interest on interest.
© 2015 Pearson Education, Ltd.

26. return2015 = principle2014 × (1 + 0.02)
7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns a 2% return ( ), what is the return after one year?
return2015 = principle2014 × ( )
Instructor: The next five slides develop the compound interest formula, using the example of a 2% investment. This is not covered in the textbook.
© 2015 Pearson Education, Ltd.

27. return2015 = principle2014 × (1 + 0.02) × (1 + 0.02)
7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns a 2% return ( ), what is the return after two years?
return2015 = principle2014 × ( ) × ( )
= principle2014 × ( )2
© 2015 Pearson Education, Ltd.

28. return2064 = principle2014 × (1 + 0.02)50
7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns a 2% return ( ), what is the return after 50 years?
return2064 = principle2014 × ( )50
© 2015 Pearson Education, Ltd.

29. Compound growth formula: returnt+n = principlet × (1 + g)n
7A.6 Calculating Average (Compound) Growth Rates
Compound growth formula:
returnt+n = principlet × (1 + g)n
where t = starting year
g = growth rate
n = number of years
© 2015 Pearson Education, Ltd.

30. 7A.6 Calculating Average (Compound) Growth Rates
We can rewrite the compound growth equation for the average annual growth rate, g:
Instructor: This slide shows the derivation of the compound growth rate.
© 2015 Pearson Education, Ltd.

31. Example: U.S. GDP growth from 1960 to 2010:
7A.6 Calculating Average (Compound) Growth Rates
Example: U.S. GDP growth from 1960 to 2010:
GDP2010 = GDP2010 × (1 + g)50
41,365 = 15,398 × (1 + g)50
g = /50 – 1 = (1 + g)50
Instructor: The top line is the definition of compound growth. The remaining lines are the calculation of the compound growth rate for U.S. GDP.
© 2015 Pearson Education, Ltd.