LECTURE 13: ECONOMIC GROWTH
13.1 THE SOLOW GROWTH MODEL
Robert Solow won Nobel Prize for contributions to the study of economic growth. It is a
major paradigm which is widely used in policy making and a benchmark against which most
recent growth theories are compared. The Solow Growth Model is designed to show how
growth in the capital stock, growth in the labour force, and advances in technology interact in
an economy, and how they affect a nation’s total output of goods and services.
How Solow Model Is Different: Assumptions
K is no longer fixed: investment causes it to grow, depreciation causes it to
shrink. L is no longer fixed: population growth causes it to grow.
The consumption function is simpler.
• No G or T (only to simplify presentation; we can still do fiscal policy experiments)
• Cosmetic differences.
13.2 THE PRODUCTION FUNCTION
Let’s analyze the supply and demand for goods, and see how much output is produced at any
given time and how this output is allocated among alternative uses. The production function
represents the transformation of inputs (labour (L), capital (K), and production technology)
into outputs (final goods and services for a certain time period).
In aggregate terms: Y = F (K,
L) y = Y/L = output per worker
k = K/L = capital per worker
Assume constant returns to
scale: zY = F (zK, zL) for any z
>0
Pick z = 1/L. Then
Y/L = F
(K/L, 1) y =
F (k, 1)
y = f (k)
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Where f (k) = F (k, 1)
13.4 THE NATIONAL INCOME IDENTITY
Y = C + I (remember, no G)
In “per worker” terms: y = c + i
where c = C/L and i = I/L
13.4 THE CONSUMPTION FUNCTION
s = the saving rate, the fraction of income that is saved (s is an exogenous parameter).
Note: s is the only lowercase variable that is not equal to its uppercase version divided by L.
Consumption function: c = (1–s) y (per worker)
13.5 SAVING AND INVESTMENT
Saving (per worker) = sy
National income identity is y = c + i
Rearrange to get: i = y – c = sy (investment = saving)
Using the results above, i = sy = sf (k)
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13.6 CAPITAL ACCUMULATION
Investment makes the capital stock bigger, depreciation makes it
smaller. Change in capital stock= investment – depreciation
∆k = i – δk
Since i = sf (k), this becomes: ∆k = s f (k) – δk
13.7 THE EQUATION OF MOTION FOR k
The equation of motion for k can be written
as: ∆k = s f (k) – δk
It is the Solow model’s central equation. It determines behavior of capital over time which, in
turn, determines behavior of all of the other endogenous variables because they all depend on
k. e.g. Income per person: y = f (k), Consumption per person: c = (1–s) f (k)
13.8 THE STEADY STATE
If investment is just enough to cover depreciation, [sf (k) = δk], then capital per worker will
remain constant: ∆k = 0. This constant value, denoted k*, is called the steady state capital
stock.
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A Numerical Example
Production function (aggregate):
Y = F ( K , L ) = K ×L K=1 / 2 L1 / 2
To derive the per-worker production function, divide through by L:
Then substitute y = Y/L and k = K/L to get
y =f (k ) =k 1 / 2
Assume: s = 0.3, δ= 0.1, initial value of k = 4.0
Exercise: Solve For The Steady State
Continue to assume s = 0.3, δ= 0.1, and y = k1/2
Use the equation of motion: ∆k = s f (k) −δk to solve for the steady-state values of k, y, and c.
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Solution:
PREDICTION:
Higher s ⇒ higher k*, and since y = f (k), higher k* ⇒ higher y*.
Thus, the Solow model predicts that countries with higher rates of saving and investment will
have higher levels of capital and income per worker in the long run.
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