A identity is a equation that is true because of the way the terms are defined. Thus, when an economist says that "gross national product is equal to the sum of consumption plus investment plus government spending on goods and services plus exports minus imports," that statement is actually just one definition of how to measure GDP.
When thinking about how budget deficits affect the economy, a different identity is typically used. This identity points out that for an economy at any given time, the total quantity of funds being saved must be equal to the total quantity of funds being invested. Or to spell it out a little more fully, the U.S. economy has two sources of savings: domestic saving, and the saving that flows in from other countries. The U.S. economy also has two sources of demand for those funds: private sector investment and government borrowing. Thus, it must hold true that when government budget deficits increase, some combination of three things will happen: 1) domestic saving will rise, to supply some of the funds needed for the rise in government borrowing; 2) the inflow of savings from foreign investors will rise, to supply some of the funds needed for the rise in government borrowing; or 3) private investment will decline, because the rise in government borrowing will "crowd out" some of the funds that would otherwise have gone to the private sector.
Again, this statement is not one where different schools of economics disagree; it holds true by definition, based on the meaning of these terms. Among professional economists, those who think budget deficits should be larger, or smaller, or about the same will all agree that if deficits rise, some combination of these three consequences must and will happen--by definition.
But how much of each will happen? Jonathan Huntley of the Congressional Budget Ofﬁce lays out some evidence in "The Long-Run Effects of Federal Budget Deficits on National Saving and Private Domestic Investment," published as a working paper by the Congressional Budget Office in February. As one might expect, the question of how more government borrowing affects these three other factors is not written in stone: it will vary both according to the specific situation of the economy and according to the econometric methods being used.
That said, Huntley describes the central estimate about the long-run effects of more government borrowing based on the review of the evidence like this: For each additional dollar of government budget deficit, private saving rises by 43 cents, and the inflow of foreign capital rises by 24 cents. Thus, [e]ach additional dollar of deficit leads to a 33 cent decline in domestic investment.
The lower range of estimates is that for each additional dollar of government borrowing, private saving rises by 61 cents and the inflow of foreign capital rises by 24 cents, so private sector investment falls by 15 cents. The higher range of estimates is that for each additional dollar of government borrowing, private saving rises by 29 cents, inflows of foreign capital rise by 21 cents, and private sector investment falls by 50 cents.
It's perhaps useful to put these investment totals in context. Here's a figure on U.S. investment levels created with the ever-useful FRED website maintained by the Federal Reserve Bank of St. Louis. The blue line on the top shows gross private domestic investment (quarterly data, seasonally adjusted annual rate). The red line shows net private domestic investment. The difference between the two lines is that a certain amount of U.S. capital wears out every year, and machinery, vehicles, computers, phone systems, and so on need to be replaced. A lot of gross investment goes to replacing existing capital, and the red "net" line thus shows the addition to the capital stock each year. Notice that for a couple of quarters toward the tail end of the Great Recession, net investment turned negative--that is, gross investment wasn't high enough even to replace the existing capital.
How much effect will the currently projected deficits have on domestic investment? Say that government borrowing now that we are past the worst of the Great Depress is about 4% of GDP each year into the future (which is roughly the CBO "baseline" estimate, which is probably optimistic for the long run). Then investment would be 1.3% of GDP lower as a result of government borrowing (that is, 33% of the 4% of GDP budget deficits). The US GDP in 2014 will be about $17 trillion. So a drop of investment equal to 1.3% of GDP is a fall in actual dollars of about $221 billion in investment.
Compared with gross private investment of about $2.8 trillion, this total doesn't look especially large. But remember, most of gross investment is replacing existing capital as it wears out. Compare a decline of $221 billion in investment to the net private investment of about $600 billion, and it looks more sizable. Thus, the budget deficit is a substantial drag on how much the U.S. economy is adding--in terms of net private investment--to its capital stock each year.