Y = I+C+G, the unlikely identity (part I)

The national income identity, Y=I+C+G, is the cornerstone of Keynesian macroeconomics. It decomposes GDP, Y,  between its main expenditure components, investment (I), consumption(C ), and government expenditures (G). It captures the essence of the Keynesian mantra, namely that in the short run, output fluctuations are driven by fluctuations in those demand components. Thus, recessions are triggered by a fall in consumer or investor confidence (animal spirits), and government  spending can be adjusted to counteract these sources of fluctuations and stabilize output.

I once naively thought that this identity decomposed total spending between the spending levels of the three “institutional sectors” (households, firms and government), and at the same time stated that spending must be equal to output (and simultaneously to total income).

In fact, things are not so simple. GDP is defined as the total value of all the final goods produced on a given territory in any given period. If the left-hand side of the national income identity is the production of final goods, its left-hand side should consist of the expenditure on final goods.  But neither firms nor the government purchase any final goods. They both are producers of goods and services and by definition only purchase intermediate goods. Only households purchase final goods. Adding government purchases and firms’ investment spending to consumer spending  therefore seems inconsistent with the definition of GDP. It looks like instead we should have Y=C in a closed economy and Y = C+X-M in an open economy, where X=exports and M=imports.

Therefore, there is more than meets the eye in this identity, and while there are some good reasons why G and I are there, the way they are included in the right-hand side is not exempt from ideology.

Let us start with investment, I. When one decomposes GDP between industries, one is careful in avoiding double-counting of economic activity,  by valuing the contribution of a firm to GDP, not by its total sales, but by its value added, defined as sales minus the cost of intermediate inputs. Yet, when we add  I on the right-hand side of the national income identity, we are guilty of such double-counting.

Consider the following simple example: Robinson Crusoe lives for two periods. In period one, he gathers two coconuts on a tree. He eats one of them and plants the other one in the ground. In period 2, this coconut has grown into a tree, which produces one coconut, that Robinson gathers and eats (I assume the preceding period’s tree no longer exists in period 2). Clearly, Robinson has eaten  one coconut in each period.  For all practical purposes, the total cumulated output of this economy over the two periods is two coconuts. Yet, conventional GDP accounting states that in period 1, GDP is equal to two coconuts, with I=1 and C=1, and in period 2, GDP is equal to 1 coconut, with I=0 and C=1. Even though Robinson has eaten two coconuts over his lifetime, conventional GDP accounting considers that this economy has produced 3 coconuts over that period. There is double counting of the coconut which has been planted in period 1.

Interestingly, if that same coconut had delivered its tree in period 1, it would have been treated as an intermediate input instead of investment, and the double counting would have been avoided. Similarly, if, instead of being planted in the ground, the coconut had been stored between period 1 and period 2, it would have been counted as inventory, and double-counting would also have been avoided, as inventory depletion comes in deduction of total GDP. In such a case one would have Y=2, with C=1 and I=1 in period 1, and Y=0, with C=1 and I=-1, in period 2 (inventories are one component of investment, and in this example, investment is only made of inventories, while in the preceding example, investment consists of fixed capital formation—the period 2  tree).

It is curious that double counting is avoided for the inventory part of investment, but  not for equipment and structures. This is in fact acknowledged in national accounting, the “G” in GDP means “gross”, that is, gross of capital depreciation. If one were subtracting capital depreciation from GDP, thus computing Net Domestic Product (NDP), one would avoid double-counting: over time, any additional unit of capital would end up been fully depreciated, and therefore fully deducted from GDP. In the Robinson Crusoe example, the tree planted in period 1 should be fully depreciated in period 2, because it is no longer used thereafter. Therefore, NDP is equal to 2 in period 1 and to 0 in period 2. The accounting is identical to the case where the coconut is stored instead of planted.

Despite that NDP is a more correct measure than GDP, all the debate and policy analysis is cast in terms of GDP.  This is not without consequences.  NDP is substantially below GDP. With a typical capital output ratio equal to 3, and a typical deprecation rate of 6 % per year, NDP is 18 % below GDP. Focusing on GDP substantially biases the comparison between countries that invest a lot, like China or Singapore, with those that invest little, like the United States. A country with an investment/gdp ratio of 50 % (the current Chinese level) will settle at a steady-state capital/output ratio as high as 8, meaning that its NDP will be only half its GDP (It will likely be “dynamically inefficient”, i.e. could consume more forever by just decumulating capital). Also, the structure of economic fluctuations will change. If investment booms are protracted, GDP will look more volatile than NDP. If they are short-lived, NDP will be more volatile than GDP.

If one cares about final goods, why take investment into account at all? Indeed, one could just ignore it entirely and focus on consumption. Consumption is much less volatile than investment, which drives most fluctuations in GDP.  Macroeconomics would be little news if one looked at consumption instead of GDP.  Furthermore, consumption is all that matters for the people (it is what enters into the “utility function”).  In other words, the theoretical case for focusing on consumption is much stronger than for GDP. Since, empirically, consumption is so smooth, the gains from stabilizing it are very small.

Of course, the reason why people decided to add investment on the RHS of the national account identity, is that when there is an investment slump, people lose their jobs – resources seem to be underutilized, which is the reason why one worries about economic fluctuations in the first place. However, this is no excuse to double-count the contribution of the workers producing the investment goods.

From a pure accounting perspective, the issue is the following: To what time period should we allocate economic activity, when the cost and benefit sides of that activity occur at different dates? Shall we consider that Robinson’s investment increases GDP at the time the tree is planted, versus the time when the coconut produced by the tree is eaten? While NDP takes the first route, consumption takes the second one.

It seems that if we care about the welfare flow of the population, we should look at consumption. If we care about the level of economic activity, NDP looks more appropriate. Here, however, we run into another inconsistency.

Let us go back to our example and assume that the tree planted by Robinson now delivers two coconuts in period 2. Furthermore, let us assume that the only effort provided by Robinson is in planting coconuts – no effort is needed to gather them. In period 1, we still have Y=2, I=1 and C=1. In period 2, we now have Y=2, I=0 and C=2. GDP is equal to 2 in each period, while NDP is equal to 2 in period 1 and to 1 in period 2. This one coconut produced in period 2, is the return to “capital”, i.e. the tree, net of its own depreciation. Given that Robinson Crusoe does not plant any tree in period 2, he does nothing at all in that period – the return to labor is zero.  We consider that there is economic activity in period 2, because the tree delivers some coconuts. Yet this tree is there just because Robinson planted it in period 1. The true cause of the tree’s presence is Robinson’s planting activity in period 1. Why do we then allocate the net return of capital to period 2 instead of period 1? Surely this cannot be because that coconut is consumed in period  2; if we cared about that we would just look at Robinson’s consumption flow, not at domestic product.  Or is it because that coconut is produced in period 2? So is the coconut which was deducted for depreciation; after all there are two coconuts available for consumption in period 2. Shall we, then, restore GDP and forget about NDP? Then we start double-counting again. Or shall we deduct the tree’s depreciation in period 1 rather than period 2? This gets us back to looking at consumption rather than domestic product.

The example illustrates that, since capital is just past frozen labor, we may allocate its net return at any given date to the period when the investment activity which generated this capital took place.  In the preceding example, this new notion of domestic product would be equal to 3 in the first period and to 0 in the second period.  This method has the Marxist flavor of computing the labor-value of capital at any given date, and allocating it to the date when the capital was produced.  Let us call this the “labor value” domestic product (LVDP).

Only labor income at date t contributes to LVDP at that same date.  Capital income is allocated to the preceding date when the corresponding investment took place. It is reasonable to use the discount rate between the two periods to convert this capital income. It is then easy to see that the contribution of investment to LVDP at date t is simply equal to the present discounted value of the income flows generated by that investment – or, equivalently, the market value of that investment. If financial markets work well, the net marginal product of capital will be equal to the discount rate, and the market value of investment will accordingly equal to the total investment cost (things would be more complicated if there were installation costs to capital). Thus, to compute LVDP, we should just add total labor income to total investment. Calling Z the LVDP, we get

Z = wL + I

In a steady state, assuming for simplicity no economic growth, we would have I=d.K, where d is the depreciation rate of capital. Total GDP is equal to Y=wL+mK, where m is the marginal productivity of capital, also equal to d+r, where r is the real interest rate. Therefore, in steady state, we have that

Z = Y – rK

If r>0, then Z<Y, because LVDP recognizes that the benefits of investment accrue later and therefore should be discounted. If r=0, though, we are at a “golden rule” steady state and GDP and LVDP coincide. Note that LVDP coincides with GDP, not with NDP. This is because LVDP recognizes that preventing capital from depreciating today brings useful future benefits in terms of consumption.

If r>0, we in fact get that LVDP is greater than GDP in steady state. In such a case, the economy is dynamically inefficient. It accumulates too much capital. In a dynamically inefficient economy, the notion of LVDP becomes dubious because investments are imputed at their market value, but the private discount rate is not efficient in such an economy.


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