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.