A Recap of Growth Theory
The Solow Model in a Nutshell
• We will now re-do the essence of the Solow model in simple steps.
• First, understand the graph on the next page.
• Understand that sy (i.e., sf(y)) line shifts the y curve down.
• Understand that the intersection point of the sy line and the δ k line will create the Steady State point.
The steady state Investment
and depreciation
Capital per worker, k
s f(k)
δk
k*
3
The steady state
• What does that mean algebraically? • δ k = sy • or, s/δ = k/y • This is the basic Solow result.
– Typically, you will be give the values of s and δ and you will need to find out the value of k.
– Example: s = 0.3; δ = 0.1; so 3 = k/y.
How to Find Steady State Values
• Now you need to know what the production function (y) is. Otherwise, you cannot solve for 3 = k/y because there are two unknowns in one equation.
• If you use a simple production function: • y = k½
• You now have: 3 = k/ k½
• Now simplify.
How to Find Steady State Values
• 3 = k/k½
• Or, 3 = (k k½)/(k½ k½) • Or, 3 = k½
• Or k = 9; so y = 3. • So per capita savings is 30 percent of y, or 30%
of 3 = 0.9 • So per capita consumption is 70 percent of y,
or 70% of 3 = 2.1
How to Find Steady State Values • If you understood the example above, you can
easily generalize to the case where there is population growth and where there is human capital growth. Simply replace δ with (δ + n + g).
• Now you need to know δ , n, g and s to find the values.
How to Find Steady State Values
• Further complications arise when the production function is a bit more complicated such as:
• We have worked through the algebra and shown that in this case the steady state value of k becomes:
How to Find Steady State Values
Solow Model Steady State Formulas
• Or,
Here Ɵ = s/(δ+n+g). Given the values of s, n, δ, A and α, we can find steady state values of per capita GDP and per capita capital stock.
A note about the nth root
• In the Solow model, sometimes you come across a problem like this:
• y = kα
• Value of y is known. What is k? • Example: 10 = k0.3
• Solution: take (1/0.3)th root on both sides to “free” k from its index.
• 10(1/0.3) = k; or k = 2154.27
Why Does Steady State Happen?
• Solow model predicts that steady state is inevitable.
• Note the meaning of steady state: – With n = g = 0, steady state Y (uppercase Y, GDP) is
a fixed number. – With g = 0, steady state Y is a growing number
even at steady state (grows at n%). – With n and g ≠ 0, steady state Y is a growing
number even at steady state:(grows at (n + g)%.
Why Does Steady State Happen? If the sf(y) was not bending down, there would be no steady state, no matter what the values of n and g are. So the next question is: why does sf(y) bend down?
The answer is that, it bends down if the production function is Cobb-Douglas – which looks like
Avoiding the Steady State
• If α = 1, then the line does bend. There will be no intersection with the depreciation line, hence no steady state.
• If we make optimistic assumptions about technology, then α may be 1, and then the future generations will not face a steady state.
• That’s the Solow model in a nutshell.
How to Find Golden Rule Savings Rate
The golden rule saving rate is given by: sg/(δ+n+g) = k/y where (δ+n+g) = MPK or, sg/MPK= k/y or, sg = (MPK)*k/y. Note that this is also the share of capital in national income.
How to Find Golden Rule Savings Rate
There is a quick way to find what the golden role savings rate is: it is the exponent of k in the production function!
Suppose the production function is: y = Akα
Then MPK = ∂y/∂y = Aαkα-1
Giving us: sg = MPK*k/y = [Aαkα-1]*k/y = αAkα/y = α
Basic Message of All Growth Theories
• High savings rate is a good thing for poor countries.
• The government can save too by increasing taxes or by reducing government expenditure. Remember that Solow model is about the very long run. It does not address short run issues of unemployment and recession.
• Low population growth is a good thing. • The most important factor in the long run is the
rate of growth of technology.
Basic Message of All Growth Theories
• Solow model gives us a fairly clear picture of where a country is headed.
• It also offers some hope to poor countries: if Poorland has the same savings rate, the same population growth rate, the same tech growth rate, the same depreciation rate and the same production function, Poorland will eventually catch up with Richland (“the Convergence Hypothesis”) provided they have the same human capital index.
• From the perspective of development economics, then, growth theories tell us where the focus should be.
• The focus should be on human capital, technology and savings.
A Recap of Growth Theory
The Solow Model in a Nutshell
The steady state
The steady state
How to Find Steady State Values
How to Find Steady State Values
How to Find Steady State Values
How to Find Steady State Values
How to Find Steady State Values
Solow Model Steady State Formulas
A note about the nth root
Why Does Steady State Happen?
Why Does Steady State Happen?
Avoiding the Steady State
How to Find Golden Rule Savings Rate
How to Find Golden Rule Savings Rate
Basic Message of All Growth Theories
Basic Message of All Growth Theories
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