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Unformatted text preview: The Solow Model
Prof. Lutz Hendricks
Econ499 January 20, 2011 1 / 37 The Solow Model In the production model capital is exogenous.
We learn how much capital matters, but we cannot learn why some
countries lack capital.
We need a model with capital accumulation (investment, saving).
That also answers the question: Does capital drive longrun growth?
We add just one piece to the production model:
an equation that describes how capital is accumulated over time
through saving. 2 / 37 Model Elements
The world goes on forever.
Time is indexed by the continuous variable t . There is one good which is produced from capital K and labor L.
The aggregate production function is
Y (t ) = F [K (t ) , L (t ) , A (t )]
= K (t ) α [A (t ) L (t )] (1) 1 −α A is an index of the state of ”technology” (anything that makes people
more productive over time).
A grows over time for reasons that are not modeled (a major
shortcoming of the model). 3 / 37 Model Elements L grows over time at rate n:
L (t ) = L (0) e n t Normalize L (0) = 1
Why can I do that? Output is divided between consumption and gross investment:
Y (t ) = C (t ) + I (t ) (2) 4 / 37 Model Elements Investment contributes to the capital stock:
˙
K (t ) = I (t ) − δ K (t ) (3) ˙
K (t ) = dK (t ) /dt is the time derivative of K (t ).
the change in K per “period”. δ is the rate of depreciation. 5 / 37 Model Elements This is a closed economy. Saving equals investment: S (t ) = I (t ).
Note: All of the above is simply a description of the production
technology. Nothing has been said about how people behave.
People make two fundamental choices (in macro!):
1 How much to save / consume. 2 How much to work. 6 / 37 Choices Work: we assume L is ﬁxed.
Consumption / saving: We assume that people save a ﬁxed fraction
of income:
Ct = (1 − s ) Yt
(4)
Equivalently:
It = sYt (5) 7 / 37 Model Summary
1 CobbDouglas production function
Y (t ) = K (t )α [A (t ) L (t )]1−α 2 (6) Law of motion for capital:
˙
K (t ) = I (t ) − δ K (t ) 3 Constant population growth: L (t ) = L (0) e nt . 4 Constant productivity growth: A (t ) = A (0) e γ t . For now: γ = 0. 5 (7) Constant saving rate: I (t ) = s Y (t ). 8 / 37 Solving the Model Even this simple model cannot be ”solved” algebraically.
That is, we cannot write the endogenous variables as functions of the
parameters.
This is almost never possible in dynamic models.
Dynamic means: there are many time periods.
All interesting macro models are dynamic. What we can do is
1
2 graph the model and trace out qualitatively what happens over time.
solve the model for the longrun values of the endogenous variables
(e.g. Kt as t → ∞). 9 / 37 The Solow Diagram We condense the model into a single equation in K .
It will be a dynamic equation that tells us how K changes over time
as a function of Kt .
Then we graph the equation. 10 / 37 The Solow Equation
Start from the law of motion:
˙
K (t ) = I (t ) − δ K (t ) (8) Impose constant saving:
˙
K (t ) = s Y (t ) − δ K (t ) (9) Impose the production function:
˙
K = sK α [AL]1−α − δ K (10) Intuition ...
11 / 37 Per capita growth
We express everything in per capita terms. E.g., y = Y /L, etc.
Output per capita is derived from Y = K α [AL]1−α :
y = ( K / L ) α A 1 −α (11) Let’s ignore technical change for now and set A constant.
Now we have
or ˙
K /L = sA1−α (K /L)α − δ K /L (12) ˙
K /L = sA1−α k α − δ k (13) 12 / 37 The law of motion for capital ˙
˙
Claim: k = K /L − nk . The law of motion can then be written as
˙
k = sA1−α k α − (n + δ )k (14) Intuition:
Suppose you invest nothing (s = 0). Then K drops by δ each period
due to depreciation.
K /L declines even more because the number of people increases by n
each period. 13 / 37 Proof of the law of motion ˙
K
Y
= s −δ
K
K
k = K /L
˙
˙
˙
k
KL
=−
k
KL
y
= s −δ −n
k (15)
(16)
(17)
(18) 14 / 37 Digression: What modern macro would do Modern macro would replace the constant saving rate with an
optimizing household.
Households maximize utility of consumption, summed over all dates.
They choose time paths of c (t ) and K (t ).
The saving rate would be endogenous and depend on
the interest rate (marginal product of K )
productivity
population growth
expectations of all future variables. What do we gain from this complication? 15 / 37 Factor prices Assume that (K , L) are paid their marginal products:
q = ∂ F /∂ K (19) w = ∂ F /∂ L (20) q is the rental price of K , it is not the interest rate.
What is an interest rate? 16 / 37 The Interest Rate Deﬁnition
The interest rate answers the question:
If I give up 1 unit of c today, how much more can I consume tomorrow?
What is the interest rate in the Solow model?
Rent 1 unit of c to the ﬁrm at t .
At t + 1 receive:
1
2 qt +1 in rental income;
1 − δ units of undepreciated capital. The interest rate is therefore: 1 + rt +1 = qt +1 + 1 − δ . 17 / 37 Analyzing the Solow Model What are the properties of the Solow model?
Why do economies grow over time?
Does the economy settle down in the longrun?
What are the longrun and shortrun eﬀects of changes in behavior?
To answer that: Plot the law of motion for k :
˙
k ( t ) = s k ( t ) α A 1 −α − ( n + δ ) k ( t )
y (t ) 18 / 37 The Solow diagram The capital stock converges over time to the steady state, k ∗ .
19 / 37 The steady state
Deﬁnition
A steady state is a situation where all variables are constant over time (in
per capita terms).
In the Solow model:
˙
Capital per worker is constant: k = 0.
The steady state capital stock solves:
sf (k ∗ ) = (n + δ ) k ∗ (21) (Intuition?) The graph shows that there is a unique steady state (because we
assumed that f has nice properties).
Parameters that determine the steady state: n, δ , s and parameters of
f.
20 / 37 Comparative statics (or dynamics)
What happens if households save more? Plot the time paths of output and interest rates.
21 / 37 Reality Check The model says: more investment (or lower consumption) generates
a period of faster growth.
Isn’t everybody saying: the U.S. is in a recession (slow growth)
because consumption is too low?
How does the contradiction get resolved?
Where is the eﬀect of lower consumption demand in the Solow model?
Where is the demand side anyway? 22 / 37 What happens if there is a baby boom? 23 / 37 Economic Growth Why do countries grow?
In the Solow model:
Growth can only occur along a transition path.
There is no longrun growth in GDP per worker (y = Y /L). But growth slows as the economy approaches the steady state.
To see this, write the law of motion for k as
˙
k /k = g (k ) = sy /k − (n + δ ) 24 / 37 Economic Growth 25 / 37 Why does investment not sustain growth? The problem is the diminishing MPK .
Giving up one unit of C today yields MPK − δ in additional output
tomorrow.
As k grows, MPK eventually falls below δ :
Additional investment no longer even pays for its own depreciation. Sustained growth through capital accumulation requires that MPK
stays above δ , even as k grows without bounds. 26 / 37 Technical change To sustain longrun growth of y the Solow model requires technical
change.
Technical change is modeled as shifting the production function up.
Productivity grows: g (A) > 0. Modern models treat A as the product of innovation.
Here: A is exogenous.
Assume that technical change is labor augmenting: Y = F [K , A L].
Otherwise, the model is not consistent with the Kaldor facts (not
obvious, but true). 27 / 37 Law of motion with technical change
Production function Y = K α (AL)1−α
˙
Law of motion (unchanged): K = sY − δ K
or
g ( K ) = s Y /K − δ
= s y /k − δ ¯
Consider variables divided by AL: y = Y / AL, k = K / AL, etc.
¯
These will be constant in ”steady state”. By the growth rate rule:
¯
g k = g ( K ) − g ( A) − n = s y /k − g (A) − n − δ
¯¯ This is the key equation for the Solow model with technical change:
¯
¯
d k /dt = s y − (n + δ + g (A)) k
¯
(22) 28 / 37 What has changed? The model with technical change looks exactly like the previous model,
except:
1 All variables are ”detrended” (divided) by AL. 2 The steady state has per capita variables growing at rate g (A). 3 The law of motion contains an additional g (A) term.
¯¯
The model has a steady state in the ”detrended” variables k , y .
It has a balanced growth path in per capita variables (k , y ).
Deﬁnition: A balanced growth path is a situation in which all variables
grow at constant rates. 29 / 37 The Solow diagram This is essentially the same diagram as without technical change, except:
variables are detrended.
an additional g term appears in the straight line.
30 / 37 Comparative statics: higher saving rate The Solow diagram is familiar: 31 / 37 Transitional dynamics 32 / 37 The Principle of Transition Dynamics Fact
In the Solow model, the farther away the economy is from its steady state,
the faster it grows (or shrinks)
What is the intuition? 33 / 37 Policies have level eﬀects A key implication of the Solow model: Policies, such as taxes, do not
aﬀect the longrun growth rate.
The growth rate rises on the transition to the new steady state, then levels
oﬀ to g (A). 34 / 37 Important Points The Solow model reveals how choices (saving, fertility) aﬀect capital
and output (levels and growth).
Capital cannot sustain longrun growth.
the reason: diminishing returns Therefore policies have level eﬀects.
In the short run: countries grow fast when they are far below their
steady states.
In the long run: growth is determined by productivity improvements. 35 / 37 Final Example Modify the Solow model by assuming that the production function is given
by
Yt = AKtα L1−α − X
t
1 How does the Solow diagram change? 2 How many steady states are there? 3 Which ones are stable? 36 / 37 Reading Jones, Introduction to Economic Growth, ch. 2 37 / 37 ...
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 Spring '11
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