Download Lecture 1: Cross-country income differences 1 National accounts

SIMON FRASER UNIVERSITY
BURNABY
BRITISH COLUMBIA
noindent Paul Klein
Office: WMC 3635
Phone: TBA
Email: [email protected]
URL: http://paulklein.ca/newsite/teaching/305.php
Economics 305
Intermediate Macroeconomic Theory
Fall 2013
Lecture 1: Cross-country income differences
1
National accounts
When we study aggregate economic activity, our main source of empirical data are national
accounts. The concepts that you will find in national accounts do not necessarily mean what
you think they mean. Also, we will see why the gross domestic product is the same whether
you look at the income or the expenditure side.
By the way, why “gross”? Because depreciation, or depletion of natural resources, is not
deducted. Suppose we have a sandwich that we made yesterday. Today we eat it and do
nothing else. Today’s gross output is one sandwich. Today’s net output is zero. Nobody ever
talks about net output. Perhaps we should. On thing is clear—we should not maximize gross
national happiness as recommended by Jigme Singye Wangchuck (then king of Bhutan) in 1972.
Presumably gross happiness means that misery is not deducted. Given that definition, what
would maximization of gross happiness entail?
1
National accounts (fictitious)
Expenditure
Income
Private consumption
Gov’t consumption
Investment
Net exports
50
25
23
2
Sum (GDP)
100
Wages and salaries (incl. SIC)
Interest payments, rents, royalties, profits
65
35
100
The reason the two sides are equal is that every transaction is recorded twice, once on the active
(“debit”) side and once on the passive (“credit”) side. According to the principles of double
Italian book keeping (which is much more honest than it sounds), every transaction has two
aspects: where the money is coming from (credit) and what it is used for (debit). A fisherman
catches a fish and sells it. The very same transaction is recorded both a purchase of fish (debit)
and as proprietor’s income (credit; but is it wages or profits?). Of course the amounts are the
same.
Factor cost or market prices? “Market prices” means that we include indirect taxes such as
GST and PST. “Factor cost” means that we deduct them.
Gross domestic product or gross national product (GNP)? GDP is produced using factors of
production employed on the territory of a given country. GNP is generated using factors of
production owned by the residents of a given country. GNI (gross national income) is a synonym
for GNP (I believe). Incidentally, the difference between GDP and GNP equals the difference
between net exports and the current account of the balance of payments.
In 2010, Ireland’s GDP was €156487 million. Its GNP was €130202 million. Source: Central
Statistics Office. That’s about 83 percent.
It is often said that Canada’s trade is 51 percent of its output. In what sense is that true? By
definition, the trade share is (exports+imports)/GDP. But the use of word “share” suggests
that we could not trade more than we produce. That is, however, not the case.
According to Statistics Singapore, Singapore’s GDP was (measured in 2005 prices) 305228
2
million Singapore dollars. The corresponding figure for exports was 689133. 226 percent. How
is that possible?
Finally, notice this. According to Statistics Sweden, Sweden’s tax revenue/GDP ratio was 42
percent in 2012. Meanwhile, here are some numbers from the national accounts of that year.
National account numbers from Sweden 2012
GDP Gov’t consumption Gov’t investment
3561903
956482
116418
Unit: millions of Swedish kronor. Source: Statistics Sweden.
Government consumption plus investment amounted to about 30 percent of GDP, not 42 percent. What accounts for the large difference?
2
Facts
Differences across countries in output per head (or per worker, or per hour worked) are vast.
According to Caselli (2004), Canada is in the 90th percentile among countries, Togo is in the
10th percentile. The ratio of Canada’s to Togo’s GDP per head is about 20. According to
Heston et al. (2011), that ratio is about 30. Meanwhile, again according to Heston et al.
(2011), the ratio of the 90th percentile to the median is about 4.5.
By comparison, the ratio of the 90th to the 50th income percentile among Canadian households
is just two.
How can these enormous differences be explained? Less ambitiously, what fraction of this
difference depends on differences in measurable factor inputs and what fraction depends on a
residual that we call “total factor productivity” (TFP)?
We will investigate cross-country output differences from the point of view of Solow’s growth
model, which will serve as our vehicle for decomposing income ratios by factor inputs and TFP.
It will also have implications for the relationship between investment rates and levels of GDP
per head, and we will use this fact to confront theory with data.
3
3
A digression on continuous time
Jones consistently treats time as a continuous variable. So we had better learn to understand
what that means and how it works. Provided that you know elementary calculus, this is mainly
a matter of establishing notation.
The most important piece of notation is the overdot, which we use to indicate a derivative with
respect to time. For any function of time x(t), we define
ẋ(t) :=
d
x(t).
dt
For example, if x(t) = t, then ẋ(t) = 1 for all t.
Another important idea is the rate of growth. By that I don’t mean the time derivative ẋ(t), I
mean the percentage growth rate, i.e. the continuous–time counterpart of
xt+1 − xt
.
xt
How should we define this in continuous time? I suggest letting time be discrete (t = 0, 1, 2 and
so on) but to see what happens as the time interval between periods gets shorter and shorter.
To make this idea as concrete as possible, suppose we have a bank that offers an infinite sequence
of savings accounts. Each account pays an annual interest rate of r, but each account differs
with respect to how often interest payments are capitalized (added to the balance). Account
1 capitalizes each year, account 2 capitalizes twice every year, etc. More generally, account k
capitalizes k times per year.
When k → ∞, we have instant (immediate) capitalization. The behaviour over time of the
balance a savings account with instant capitalization will be our definition of what it means to
have an annual growth rate of r in continuous time. But how does it behave? Let time t be
measured in years and define ∆t := 1/k.
Apparently
xt+∆t = (1 + r∆t)xt
which implies that
xt+∆t − xt
= rxt .
∆t
Now let k → ∞ so that ∆t → 0. Then we get
ẋ(t) = rx(t)
4
or, put differently,
ẋ(t)
= r.
x(t)
(1)
Motivated by this, we say that x(t) has the growth rate of r precisely if (1) holds.
Notice that, rather conveniently,
ẋ(t)
d
= [ln x(t)].
x(t)
dt
Moreover, if x(t) = er·t , then the growth rate of x(t) is constant and equal to r.
Finally, recall that the logarithm function has the following property: ln(xy) = ln x + ln y. This
implies the following important result. Let x(t) grow at rate γ x and let y(t) grow at rate γ y .
Then the product z(t) := x(t) · y(t) grows at rate γ z = γ x + γ y .
4
The Solow model in continuous time
Again we start in discrete time and then shorten the time periods ad infinitum. Suppose real
output Yt is produced according to
Yt = F (Kt , Lt )
where Kt is the stock of capital and Lt is the labour input. Ignoring growth in technology or
population so that Lt = L, the Solow model says that
Kt+∆t = sF (Kt , L)∆t + (1 − δ∆t)Kt
where s is an exogenously given constant. Dividing by ∆t and letting ∆t → 0, we get
K̇(t) = sF (K(t), L) − δK(t).
In the long run, if there are diminishing returns to capital and the initial capital stock is strictly
positive, the capital stock will converge to a strictly positive constant K ∗ such that
sF (K ∗ , L) = δK ∗
We call this the steady state. See Figure 1.
5
6
δK
sF (K, L)
-
-
K
∗
-
K
Figure 1: Dynamics in the Solow model
5
Constant returns to scale in the Solow model
We will assume that F exhibits constant returns to scale, i.e. that, for any λ > 0, K ≥ 0 and
L ≥ 0, we have
F (λK, λL) = λF (K, L).
This implies — provided the derivatives involved exist — that if factors of production are paid
their marginal products, then income exhausts output, i.e.
F (K, L) =
∂F (K, L)
∂F (K, L)
·K +
· L.
∂K
∂L
Example: Cobb–Douglas.
Suppose Y = F (K, L) = K α L1−α where 0 ≤ α ≤ 1. Then
∂F (K, L)
Y
=α
∂K
K
and
∂F (K, L)
Y
= (1 − α) .
∂L
L
6
Clearly income exhausts output when factors are paid there marginal products in
this case.
The assumption of constant returns to scale greatly simplifies the treatment of growth either
in population or (labour) productivity or both. The two cases are formally similar, so suppose
there is just population growth and that it is constant at rate n. Then
L̇(t) = nL(t).
Then we have
K̇(t) = sF (K(t), L(t)) − δK(t)
and if we define k(t) = K(t)/L(t) we get
k̇ = K̇ ·
k
− nk
K
and it follows that
k̇(t) = sF (K(t)/L(t), 1) − (δ + n)k(t).
Defining f (k) := F (k, 1) we have, finally,
k̇(t) = sf (k(t)) − (δ + n)k(t).
6
Long–run behaviour of capital per worker
The Solow model implies that the capital/labour ratio converges to a fixed level. Let’s compute
the long–run level of capital/labour ratio! Suppose
f (k) = Ak θ
where A is some constant and θ is capital’s share of income. By definition of k ∗ , k̇(t) = 0 when
k(t) = k ∗ and hence
sA(k ∗ )θ = (δ + n)k ∗ .
Hence
(
∗
k =
sA
δ+n
7
1
) 1−θ
and the long–run level of output per worker is
(
1
∗
1−θ
y =A
s
δ+n
θ
) 1−θ
.
Notice that the long–run level of output per worker depends on the parameter A as well as the
investment rate s and the depreciation rate δ.
7
The investment rate and long–run output per worker
As we have seen, the Solow model has very definite implications for the relationship between
the investment rate and long–run output per worker. Taking long–run growth into account will
not change this implication. Specifically, we have
∆ ln y
θ
=
.
∆ ln s
1−θ
That is to say, if θ = 1/3 (as suggested by national accounts), then our theory implies that
∆ ln y
1
= .
∆ ln s
2
As documented in, among other places, Mankiw et al. (1992), this isn’t the case. They conclude
that a capital share of 2/3 would improve the empirical fit of the Solow model.
Using statistical techniques (multiple regression) it is possible to estimate the slope of the
relationship between ln y and ln s across locations at a particular point in time. Actually,
Mankiw et al. (1992) use a panel, i.e. observations from different places and times.
Without getting bogged down into details, we may approximate the slope in question by taking
the most recent observation of (log) output per person and plotting it against an average of (log)
investment rates over as long a period as we can find. Then you can use eyeball econometrics
to draw a line through the data and measure its slope.
The investment varies quite a bit over time, that’s why it’s appropriate to take an average.
On the other hand, it is not appropriate to take an average of output per head, since it has a
trend. If one is ambitious, what one can do instead is to estimate a growth rate γ (assumed
common to all countries in the sample) and divide all output figures by (1 + γ)t . The result
can then be averaged over time.
8
How do we measure the investment rate? Well, investment is the sum of three things:
• Government gross fixed capital formation
• Business gross fixed capital formation
• Net increase in inventories
What about the following?
• Expenditure on durable consumption goods, including the building and maintenance of
owner–occupied housing
• Some appropriate fraction of expenditure semi-durable consumption goods
Although conceptually these constitute investment, they do not augment that part of the capital
stock that is used to produce measured GDP. So I suggest excluding them from the measured
investment rate.
To find the investment rate, one of course divides investment by output. Real or nominal? It
is mildly interesting to consider why it makes a difference.
There is an upward trend in all price indices because of inflation. Thus the deflator used to
convert nominal output to real output increases over time as does the deflator used to convert
nominal investment into real investment.
It it tempting to think that these two deflators should have the same trend — they should both
grow at “the” rate of inflation. However, this is rather far from being the case. As documented
by Greenwood et al. (1997), investment goods have become relatively cheaper over time. Since
1950, the relative price of equipment has gone down by two thirds in the United States.
But what measure is better, nominal ratios or real ratios? That depends on the purpose of
the analysis. Here we are interested in what determines the long–run level of output in a
cross–section of countries. What determines this is not (directly) the amount of expenditure on
investment goods in each country (measured in terms of consumption goods), but how much
capital is actually purchased.
There is strong evidence that the relative price of investment goods in terms of consumption
goods varies a lot not just over time but across countries, especially when poor countries are
9
in the sample. Restuccia and Urrutia (2001) find that the relative price of investment goods
(in terms of consumption goods) differ across countries from about 1 in rich countries (this is
a normalization) to about 6 in very poor countries.
Do these higher prices translate into lower rates of investment in poor countries? Interestingly,
it turns out that the ratio of investment expenditure to GDP doesn’t differ all that much across
countries. The range is 10-30 percent.
More to the point, the relationship between income per capita and the ratio of investment
expenditure to GDP is weak. Belgium has a similar investment expenditure to GDP ratio as
Zambia. Japan is similar to Togo in this respect.
Howeber, since capital goods are relatively more expensive in poor countries, then each dollar’s
worth of forgone consumption buys less productive capital in Senegal than in Sweden.
We therefore expect that there should be a tighter relationship between investment rates and
output per head if we measure investment as the amount of productive capital accumulated
rather than just as expenditure. And that is indeed the case.
As it turns out, both the Mankiw et al. (1992) and the Restuccia and Urrutia (2001) data
suggest that the relevant slope coefficient is about 2 (not 1/2). Thus we can back out the value
of θ via
θ
≈2
1−θ
from which it follows that
2
θ≈ .
3
What are we to make of this? Human capital? That is the subject of the next lecture.
10
NOTES TO STUDENTS
1. Equations involving ratios should look like this:
x+
and not like this:
z
a+b
=
y
c+d
x + a+b = z
y
c+d
/ but I don’t recommend it. The following are
2. The number 0 may be written like this: 0,
not appropriate: Ø (a Danish/Norwegian letter) and ∅ (the empty set).
11
References
Caselli, F. (2004). Accounting for cross-country income differences. Draft for a chapter in
Handbook of Economic Growth.
Greenwood, J., Z. Hercowitz, and P. Krusell (1997). Long-run implications of investmentspecific technological change. American Economic Review 87 (3), 342–362.
Heston, A., R. Summers, and B. Aten (2011). Penn world table version 7.0. Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania.
Mankiw, G., D. Romer, and D. Weil (1992). A contribution to the empirics of economic growth.
Quarterly Journal of Economics 107, 407–438.
Restuccia, D. and C. Urrutia (2001). Relative prices and investment rates. Journal of Monetary
Economics 47 (1), 93–121.
12