The government in the Solow model

In Chapter 1 we summed up the leakages from and the injections into the circular flow of income in the equation S — I + T — G + IM — EX = 0. Rearranging this into

Arranged this way, the circular flow identity reveals that national saving is either invested at home or abroad.

Domestic investment

Investment abroad

Private saving

Public saving

Total private investment

National saving reveals a more general correspondence than the simple I = S equation used in the basic Solow model. The complete circular flow identity implies that all national saving, both private and public, equals total private investment, at home and abroad. When discussing economic growth in Chapter 9 we ignored the government sector and international trade (setting T = G = IM = EX = 0), thus ending up with the equality between private saving and investment S = I. Let us now keep the government in the equation while still leaving out the foreign sector (the role of foreign investment will be discussed in the next section). Investment is then determined by

So investment may be financed by private and public saving, since T — G is the government budget surplus, or government saving. Assuming that individuals save a constant fraction s of disposable income, S = s(Y — T), where s = 1 — c, we obtain

Now recall from the last chapter that the capital stock K changes if investment exceeds depreciation, aK = I — 8K. Substituting (10.1) into this equation, making use of the production function Y = F(K, L), and rearranging terms gives aK = sF(K, L) + (1 — s)T — G — SK

The first three terms on the right-hand side represent national savings (which equals investment). The last term is depreciation. Following the line of argument employed in Chapter 9, we may determine the steady state (in which AK = 0) graphically (see Figure 10.1). SK is a straight line through the origin. National savings is composed of sF(K, L), the broken dark blue curve that is proportional to the production function, and the terms (1 — s)T — G, which bear on the vertical position of the national savings line.

Figure 10.1 reveals why high government spending is considered so harmful for the longer-run prospects of the economy. A rise in government spending shifts the savings line down, reducing national savings and investment at any level of K, reducing the steady-state capital stock and steady-state income.

The obvious reverse side of this is that taxes do exactly the opposite. As they rise, national savings and investment increases and steady-state income moves higher. But then why do economists not fervently recommend tax


Figure 10.1 The no-government (G = T = 0) steady state features the capital stock K* = G = 0 and income Y*. Raising government spending and driving the budget into deficit shifts the savings line down, lowering the steady-state levels of K and Y. Taxes operate as involuntary savings. As they rise, the savings line shifts up and the steady-state levels of both K and Y rise.

Figure 10.1 The no-government (G = T = 0) steady state features the capital stock K* = G = 0 and income Y*. Raising government spending and driving the budget into deficit shifts the savings line down, lowering the steady-state levels of K and Y. Taxes operate as involuntary savings. As they rise, the savings line shifts up and the steady-state levels of both K and Y rise.

increases? There are a number of reasons - some more of an economic nature, and some more political:

■ Remember that the variable we would ultimately like to maximize is consumption. Just as the Golden Rule gave us an optimal private savings rate in the last chapter's basic Solow model, similar reasoning yields an optimal, golden national savings rate in the current extended model with government. If national savings is already at the level suggested by the Golden Rule, tax rises would be detrimental to steady-state consumption.

■ Even if conditions are such that a tax rise would raise steady-state consumption, its effect on current consumption is negative. This is because the current capital stock and current income are given, and higher taxes leave us with less income at our disposal. Consumption then develops according to the lower adjustment path outlined in Figure 9.16. A decision to raise taxes in order to spur national savings then involves a weighing of current consumption sacrifices against future gains. If we place less weight on future consumption compared with current consumption, it may well be rational not to raise taxes.

■ A tax increase does not only lower current potential consumption at given current potential income, as proposed by the Solow model. As we learned from our discussion of business cycles in Chapters 2-8, raising taxes will also drive the economy into a recession, driving income and consumption temporarily below their respective potential levels. This aggravates the argument advanced in the previous paragraph.

■ Governments exhibit a tendency to spend all their receipts, thus raising G whenever T rises. Raising G and T by the same amount, however, reduces investment and steady-state income. This is because a €10 billion increase in G shifts the savings line down by exactly €10 billion, while the matching €10 billion increase in T shifts the savings line up by only €8 billion (supposing s = 1 — c = 0.2). The attempt of the government to save by raising taxes leaves the private sector with less disposable income (€10 billion less). So individuals save €2 billion less. A rise in taxes, that is an increase in public savings, crowds out some private savings.

Empirical note.

Governments typically spend a rather small share of outlays on investment projects. In Germany, for example, the government invests less than 4% of its spending.This falls way short of private savings rates, which run around 25%.

The Ricardian equivalence theorem is named after British economist David Ricardo (1772-1823) who first advanced the underlying argument.

■ It is very important to note, and often overlooked, that for the above results to hold we must assume that the government only consumes and never invests. This is obviously not true as a certain share of public investment goes into roads, railways, the legal system, and education. How does this affect our argument? Suppose, government spending is composed of government consumption GC and government investment Gj, so that G k Gc + Gj. Then the capital stock changes according to aK = I + Gi - 8K

The circular flow equation I = S + T — GC — Gj can be solved for total investment, private and public, I + Gj = S + T — GC. Substituting this into equation (10.2) gives aK = S + T — GC — SK

Suppose, further, that the government routinely invests a fraction a of all government spending, so that Gj = aG and GC = (1 — a)G. Substituting GC = (1 — a)G and S = s[F(K, L) — T] into equation (10.2) gives aK = sF(K, L) + (1 — s)T — (1 — a)G — SK

The question of whether an increase in government spending that is being financed by a tax rise of equal size boosts steady-state income or not, does not have a clear-cut answer. It obviously does boost income if a > s, that is if the government invests a larger share of its spending than the private sector is prepared to save and invest out of disposable income. Then total investment, the steady-state capital stock and steady-state income all rise. A rise in G that was fully used for public investment would certainly push up steady-state income, even if accompanied by a tax increase of equal size. Note, however, that during the transition to this new, better steady state, individuals have to make do with lower disposable income and lower consumption. By contrast, matching reductions of G and T always bear short-run gains in consumption, even though the long-run, far-away options are worse.

Some economists advocate an extreme view of the crowding out of private savings by taxes that we encountered above. The Ricardian equivalence theorem maintains that government deficit spending does not affect national savings at all. In terms of Figure 10.1, no matter whether G rises, or T rises, or both rise, the savings line does not change; the government does not do anything to the steady state. The reason, according to this view, is that households realize that running a deficit and adding to the public debt today will lead to higher interest payments and eventual repayment in the future. To provide for the higher taxes that will then be needed (to provide for interest payments or repayment), individuals start saving more today. They save exactly the same amount the government overspent. The essence of this argument is that it is irrelevant whether higher government spending is financed by higher taxes or by incurring debt. In no case will it reduce national savings, but only private consumption.

The main argument advanced against Ricardian equivalence is that lives are finite. Then people have no reason to save more if they expect future generations to repay the debt. The counter argument here is that since people typi-

cally leave bequests, they obviously care about the welfare of their offspring. This should make them act as if lives would never end. If a smaller weight is placed on the utility of our children, grandchildren and so on as compared to our own utility, this weakens the Ricardian equivalence argument. Private savings may then be expected to respond to budget deficits in a Ricardian fashion, but not to the full extent of keeping national savings unchanged. This is also very much what the mixed empirical evidence on the issue seems to suggest.

But then if continuing deficit spending and growing debt is crowding out some private savings and investment, isn't this justification enough to oppose deficits and debt? Not generally - the point to emphasize is that deficit spending crowds out private investment. As we have already argued above, total investment, public and private, is only then guaranteed to fall if the deficit is caused by government consumption. If the government is running up the public debt by investing in education, infrastructure, basic research, national security, and so on, the call can be made only after comparing the returns of the government's projects with the returns of the private projects that are crowded out. Returns on the first category can be extremely high. Frequently cited examples are wars that typically make the national debt explode.

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