Lagged income and the omitted accumulating variables in a MRW production function

In the MRW production function, production per person y at a particular time is given by

y=A.k^a.h^b

where k is physical capital per person, h is human capital per person, and A, a, and b are constants. A is meant to capture the average effect of all other terms influencing production.

The dynamics of y are introduced by many authors through assuming that capital accumulates as Dk = s.y(t-1) - p.k(t-1), where s and p are constants, y(t-1) is y at time t-1, and Dk=k(t)-k(t-1). This equation allows y to be replaced by A.k(t-1)^a.h(t-1)^b, so that if Dk=0, we can solve for a value of k which means that there is no further change in y from capital accumulation. We could assume a similar expression for human capital and solve it to find a pair of (k,h) which ensures that y is unchanged forever, or only experiences random shocks which are corrected by the same adjustment processes.

So if k and h are changed - either from outside the system, for example because of foreign direct investment or immigration, or because of capital accumulation - then output adjusts instantly to the inputs. The accumulating equations are the sources of adjustment in the economy, and do not moderate the speed of output response to new inputs. There is a problem here, because one might expect some delay as output adjusts to the new inputs. New markets might have to be set up to sell the goods, or people's expertise might have to be included in the productive process.

We could enter a new factor n in the production equation to allow for the effect of all other accumulating variables:

y=A.k^a.h^b.n^c

There are many possible omitted accumulating variables, so that the constant c would vary a great deal between two different economies. Taking logs and differencing gives

Dy=a.Dlnk+b.Dlnh+c.Dlnn

If we are going to estimate in an equation, we need some proxy variable for Dlnn. It is probably unobservable and certainly difficult to observe, so finding a proxy can be awkward. But we have the lagged income term, y(t-1). This has a neat property that

correlation(Dlnn, y(t-1)) =
(Cov(Dlnn, c.lnn) + other terms which are probably less important)/ Var(Dlnn).Var(y(t-1))

So if the first covariance term is large relative to the variation in income and in the accumulation, ie if the n variable is a strong accumulator, this term will be close to unity and y(t-1) will be a good proxy for the change in lnn. This is a likely reason why the lagged income term is so helpful in growth regressions. Our explanation differs from the criticised linearisation-around-a-steady-state theory of the origin of the lagged term, or the also criticised proxy-for-technology theory of origin. Unlike them, our explanation gives a very clear reason why the lagged term is unstable.

Equipment for African research

When preparing economic research papers, there are many doh! moments where one realises that one of the assumptions or methods used much earlier was wrong, and the whole thing has to be reworked. Happily, in theoretical analysis, it is usually just a matter of adjusting a few formulae and the rest of the reworking is automatic, and in applied analysis, frequently the computer programs supplied at most universities just need a few tweaks and everything else follows.

For applied analysis without computer programs, the reworking can be painful. I know it all too well. It can multiply the time taken for a research project many times. In African universities where necessary equipment for economic and other scientific research is either missing or rationed, the resulting delays in research must be frustrating.

The problem is longstanding in Africa. In the 1930s and 1940s when several (primarily Francophone, given their educational system) African writers were producing works comparable with those of their European contemporaries, I think it is fair to say that there were few, if any, scientists from the occupied continent who could match the best of the West. I may be ignorant of African history here, but surely the lack of equipment would have seriously hindered their scientific production.

Growth theory's predictive limitations

In several major papers on economic growth, the estimates of convergence to steady states vary between two percent and eight percent each year. That means that every year, the difference between current and potential income reduces by between two percent and eight percent, where economic potential is determined by capital per person and education per person. At two percent narrowing every year, within forty years the difference should have halved, while at eight percent, the difference will halve within a decade.

The potential for many developing countries is also estimated to be higher than developed countries since their savings rate can be much higher (which provides an estimate of capital, at least once the country reaches its steady state). So why are these developing countries not already far richer than the West? They have generally had at least a decade of capitalist growth, but are still generally quite poor.

Part of the problem is with omitted variables, which account for much of the growth. Misspecification in assuming a country constant and constant autoregressive parameter in the growth equation when the true autoregressive parameter is variable across countries leads to overestimates of growth at low levels of income and underestimates at high levels, as demonstrated in one of my earlier posts. And standard estimation methods based on GMM tend to overstate growth generally in misspecified AR(1) models.

Selection of non-autoregressive parameters in GMM under misspecification

Here's an observation about the behaviour of the GMM estimates in a misspecified AR(1) model. The estimated model is

y(t) = a.y(t-1) + b.x(t) + error

while the real generating process is

y(t) = a(i).y(t-1) + b(i).x(t) + error.

The a(i) and b(i) vary across groups. I showed on 14th July that if the b(i)=b=0 for all i, then GMM selects an a estimate near the top of the range of the a(i). This post discusses what happens if the bs are not zero.

My procedure for analysis is that one takes the standard formula for an OLS estimate of the a and b parameters, then plugs in the misspecified formulae, and rearranges the formulae a wee bit to make the expressions look like weighted averages. If the b(i)s are all the same, then no matter what the behaviour of the as, the estimate for b is unbiased. If the b(i)s vary, then the estimator tends to select the parameters from the series with the highest xs, and the lowest autoregressive parameters.

I haven't worked this analysis through the GMM System and Difference estimators, but suspect that the behaviour will be the same, as the OLS is a variant of GMM. Cross-country growth theory empirics often use these GMM estimators, so the estimates will tend to be near the top of the range across countries for the lagged autoregressive parameter, and the non-autoregressive parameters will tend to come from low autoregressive parameter countries.

The next couple of sentences are more conjectural. Given that the majority of growth tends to be explained outside the augmented classical determinants of capital and education accumulation, countries whose growth is substantially explained by these determinants may be among the low growth countries with low autoregressive parameters. Thus, the non-autoregressive parameters may be near the top of the range too, and growth is likely to be overstated by the estimates.

What sort of technology promotes growth best?

The theoretical and empirical finding in many works on growth and technology is that technology is most effective in promoting growth when it is suitable for the education level in a country. Thus, sending biotechnology laboratory equipment to a country where no-one can read is unlikely to promote growth as much as sending tractors. My own current studies find that if, in a particular year, a poor country has far fewer computers per person than other countries, their subsequent growth isn't affected much. On the other hand, if they have fewer telephones in a particular year, their future growth tends to be higher than if not, other things being equal. A plausible explanation is that technology catch-up is occurring; transfers of telephone technology are usually both feasible and suitable for the education levels in a poor country, whereas computer technology transfers are often just feasible.

But there is a caution on this interpretation. A gap affects growth through its influence on transfers. Thus the causality of effect on growth is gap-->transfers-->growth. The "inappropriate for education" hypothesis states that the second stage breaks down. But in low income countries, the first stage may also break down for certain goods, for a variety of reasons.

The meaning of technology, reprise

In a post from the Monday before last I defined technology as

"physical capital with a high level of education or knowledge required in use or maintenance."

The technology level might be defined as this education or knowledge, technological transfers as changes in technology by virtue of foreign influences, and technological gaps as the difference in technological levels between two countries.

These definitions sound like they might be OK, but encounter problems when one tries to use them in a production function. For example, with the famous (MRW) production function, we have

Output per person = A*capital per person^alpha*education per person ^beta

where A, alpha, and beta are constants. The technology level is a measure of education, so presumably education per person should be decomposed into something like

Output per person = A*capital per person^alpha
*(technology-relevant education per person) ^beta1
*(non-technology-relevant education per person)^beta2

where beta1 and beta2 are new constants. But then, how would the (non-technology relevant education per person) variable be defined? And if the technology changes, how should education be repartitioned? Furthermore, if we look at the original MRW production function, it is apparent that all education helps to increase the output of physical capital, so all education could be defined as the technology level, and our definition is weak.

Similar problems arose when I tried to define technology as the part of total factor productivity due to education. After playing around with various definitions, I think the following set works and captures the main content of the theory of scientific technology transfers:

"An economy’s technology level is defined non-uniquely as a measure of the scientific processes employed when using or maintaining aggregate physical capital. A technology gap is the difference in the technology levels between two economies. Technology transfer is defined to be an inflow to an economy from another economy, whether of knowledge or physical capital or another quantity, which change the scientific processes employed in use or maintenance of physical capital."

The definitions do not overlap with definitions of other variables, so the technology level can be entered into production functions without creating problems of determining the residual parts of variables which are not captured by technology levels. They allow for neat theoretical expressions of production functions including technology which embed many other production functions in them, whether they have included technology's effects or not.

Recession or cycle?

There is lots - lots and lots - of talk in the UK at the moment of an economic recession. High oil prices are blamed for much of the problem, no doubt accurately. But will it be a recession? There is another candidate, a long term shift in the growth of the UK and other Western economies.

Recessions in the 1970s or early 1980s followed oil price hikes when the OPEC cartel raised its prices. The UK emerged from the recession following price reductions. This time, oil price increases are caused (as I mentioned in my last post) in large part by increased global demand from emerging countries whose economic size will soon be comparable with the combined size of current developed nations. This demand won't be reversed barring major global economic downturns when all countries will be in trouble, so the UK and the West will have to adjust to higher prices permanently. These prices have been associated with recessions in the past. So, the West may have to move along with slower growth for a longer period.

There will be an upswing eventually. It relies on the emerging countries wanting to buy Western goods, innovating and transferring technology to the West, and generally acting as Western countries do today. This shift takes time, so the current downturn in the UK economy may be less of a business cycle-type recession, and instead one of the longer cycles which economic historians have identified.