Sunday 4 January 2009

Income and prices as technology spread co-determinants

Here is an illustration of one of the pitfalls of measuring how technology spreads across countries. It concerns the relation of income and technology prices, which are sometimes tested as possible co-determinants of the technology level in a country, such as:

Computers per person = a*national income + b*computer cost + c*other variables

where a and b are scalar constants and c is a vector constant. National income and computer cost are collinear, their constants being connected through the Slutsky equation:

deltaQuantity/deltaPrice =
deltaQ(U fixed)/deltaPrice - quantity*deltaQuantity/deltaIncome

where delta means partial differentiation, and Q(U fixed) denotes the quantity minimising costs for a particular level of consumer satisfaction (this is the usual taught formulation of the Slutsky equation and is a bit unappetising but I do not regularly teach it so don't have any better approach - apologies). The equation means

change in quantity =
change due to substitution with other goods because of relative price changes
+ change due to lower total purchasing power as the cost rises.

In the computer example, it becomes

b = Substitution effect (which is always negative) - a*quantity.

The collinearity should theoretically be recognised and corrected for when estimating, but it isn't always. Of interest here is interpreting a result from a recent paper looking at computer and internet dissemination which finds b is small and a is large (after adjusting quantity, price, and income to be in comparable units). For b to be small, the substitution effect would have to be small. Plausibly it is in poor countries (computers and the internet are not wanted much no matter how their price varies?) and in rich countries (computers and the internet may be cheap to begin with or have no substitutes?). Additionally, a*quantity would have to be small, which means that the quantity must be small since a has been found to be large. It probably is small in developing countries, but not in developed countries.

So the observed finding of small price effect and large income effect may indicate misspecification, collinearity, or income proxying for omitted variables.

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