Y=I+C+G, the unlikely identity, part II

I now turn to the discussion of G, the level of public expenditures. We recall that since GDP is the total value of the final goods produced, only expenditures on final goods should appear on the right-hand side of the identity. Yet, the government is a producer and does not spend anything on final goods. Therefore, G stands for government production, not government consumption. In other words, G is the total value of households’ consumption of publicly provided goods. The problem is that this value is not measured, because most of those goods are provided to the households for free, or for a price which does not reflect their hedonic value. For this reason we use government consumption as a proxy for government production. We measure the output of the government by its costs.

An immediate consequence is that, contrary to the private sector, productivity growth in the public sector will not be reflected in GDP. Suppose workers can either produce 1 unit of the private consumption good or a units of a public good. In equilibrium wages should be equal to 1. If H denotes the total provision of the public good, then public expenditure equals G=H/a. If the total labor force is equal to L, then L-H/a people work in the private sector and H/a people work in the public sector. Therefore, C=L-H/a and measured GDP equals Y=C+G=L-H/a+H/a=L. Total measured GDP is therefore independent of productivity in the public sector a. When it rises, if H is unchanged, more private goods are produced for the same level of public services; GDP should therefore go up. But national accountants wrongly consider that fewer public goods are produced when the public sector cuts costs, in a way which exactly offsets the contribution of higher private consumption.

In this example the economy is in a full employment equilibrium. How do things work in a Keynesian underemployment equilibrium? Here we have to distinguish between measured GDP (Y), total income (U), and actual GDP (Z). Assume a consumption function C = m(U-T)+b, where T is the level of taxes. Assue again unit wages, and a unit price of the consumption good. We have that U=C+H/a=C+G, which decomposes total income between that of private goods producers and that of public good producers. Consequently, Y=U: total income actually matches measured GDP*. Assuming a balanced budget, we have T=H/a. Therefore U=Y=H/a+b/(1-m).  Now measured GDP unambiguously falls when a goes up, as long as H goes up less than proportionally, i.e. as long as some downsizing takes place in government. Consumption is equal to b/(1-m), and therefore actual GDP is equal to Z=b/(1-m)+uH, where u is the appropriate hedonic price, or marginal utility, of public services. Actual GDP goes up as long as the public sector is downsized less than proportionally to the raise in public sector productivity.**

These examples suggest that the way we impute government services is far from innocent. At the end of the day, official GDP growth figures make the headlines. A politician would think twice about making the public sector more efficient, if this delivers negative news in the short run, and no measured effect in the long run. The conventional expenditure approach to GDP accounting embodies a built-in bias in favor of artificially growing the public sector by reducing its efficiency, which of course benefits some interest groups such as public sector unions. Ideology can pervade not only the way models are formulated, but the very measurement of our key concepts.

NB:* This is a tricky concept of income; if public expenditure does not measure the true value of public goods, part of the wages of civil servants should be interpreted as a transfer. U would then differ from the relevant concept of pre-tax, pre-transfer income. And U would be artificially boosted by just raising taxes and government wages. What is truly relevant here is rather disposable income U-T, which determines consumption.

** It is optimal in the short run to increase H proportionally to a, because the civil servants who are laid off do not find jobs in the private sector (due to wage and price rigidity). In this case, we have no change in measured GDP.


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