Estimating the cost of stimulated jobs with an investment feedback and in the long run

From today's Krugman (Bad Faith Economics):

So as a public service, let me try to debunk some of the major antistimulus arguments that have already surfaced. Any time you hear someone reciting one of these arguments, write him or her off as a dishonest flack.

First, there’s the bogus talking point that the Obama plan will cost $275,000 per job created. Why is it bogus? Because it involves taking the cost of a plan that will extend over several years, creating millions of jobs each year, and dividing it by the jobs created in just one of those years.

...

The true cost per job of the Obama plan will probably be closer to $100,000 than $275,000 — and the net cost will be as little as $60,000 once you take into account the fact that a stronger economy means higher tax receipts.

Krugman makes that last calculation at Conscience of a Liberal (Bang for the Buck):

Consider an increase in government spending; assume that the interest rate is fixed (a good assumption right now, because interest rates are up against the zero lower bound). Then textbook analysis says that if the stimulus is dG, the increase in GDP is 1/(1 - c(1-t)) where c is the marginal propensity to consume out of income and t is the marginal tax rate. Suppose c is 0.5 and t is 1/3; then the multiplier is 1.5, which is more or less the conventional wisdom right now.

But if $100 billion in spending raises GDP by $150 billion, and the marginal tax rate is 1/3, $50 billion of the spending comes back in additional revenue. So bang for the buck — increase in GDP per dollar of added debt — is 3, not 1.5. Since the main concern about stimulus is that it will add to government debt, it’s this bang for the buck measure, rather than the multiplier, that’s relevant. And 3 sounds a lot better than 1.5.

Take this a bit further: $150 billion is about 1 percent of GDP, which Romer and Bernstein say means a million jobs; so this says $50,000 per job, which is a much better number than the critics have been throwing around (plus many more workers with full-time rather than part-time jobs).

I'm not sure where the difference between $50,000 and $60,000 per job comes from but ... no matter. I think the cost per job is larger. Here is the model that Krugman is using:

1. Y = C + I + G + NX
2. C = a + c(Y - tY)

where Y is national income (and GDP), C is consumption, I is investment, G is government spending, NX is net exports (X - M; exports minus imports), a is autonomous consumption, c is the marginal propensity to spend and t is the marginal tax rate. Plugging equation 2 into equation 1 and solving for Y gives:

3. Y = (a + I + G + NX)/(1 - c + ct)

Taking the derivative of Y with respect to G gives the government spending multiplier:

4. dY/dG = 1/(1 - c + ct)

Using Krugman's numbers, if dG = $100, c = 0.5 and t = 0.33:

4.' dY = 100/(1 - 0.5 + 0.165) = 100/0.665 = $150

And a $100 increase in government spending generates $150 in income. The $150 in income increases tax revenue (T) by $50 (T = tY) and the deficit increases by $50 instead of $100.

However, another component of this Keynesian model is the national income accounting identity:

5. C + S + tY = C + I + G + NX

Solving for I gives the investment equation:

6. I = S + (tY - G) - NX

This equation says that investment spending is equal to private savings (S) plus government savings (tY - G). If the government runs a deficit then investment spending is "crowded out" unless the capital account surplus (-NX) offsets government dissaving (i.e., unless China buys government bonds).

Now, suppose NX = 0 (i.e., China stops buying government bonds perhaps if the U.S. debt to GDP ratio gets scary and U.S. default risk becomes real). Then with dG = $100, tY - G = -$50 and investment spending and Y falls by $50. The cost per job created (using the Romer-Berntein rule of thumb of $150 billion in GDP for 1 million jobs) is back up to $100,000.

So, the relatively low cost per job created requires that foreigners keep buying government bonds (i.e., "assume that the interest rate is fixed"). Current trajectories of the debt to GDP ratio make crowding out, an increase in interest rates that reduces investment spending, more and more likely.

Crowding out is even costlier when you consider the supply side where we have the aggregate production function:

7. Y = f(K, L, Z)

where capital (K), labor (L) and technology (Z) are combined to determine long run economic prospects. Since capital is the physical manifestation of investment, if investment spending falls then capital accumulation falls and national income (GDP) falls in the long run. In the long run, with crowding out, the cost per job created is greater than $100,000.

I think.

I don't know what numbers should be plugged into equation 7, but I doubt the cost per job created would rise to $275,000 so I hope I'm not written "off as a dishonest flack."

I hope.

Update: I messed something up. If the capital account rises by $50 to suck up the increased supply of government bonds then NX will fall by $50 and the budget deficit cost of the stimulus on GDP is simply displaced from investment to net exports. So, it doesn't matter if interest rates rise or not for some form of crowding out to occur ... in this simple model.

Did I get that right?