Logic Fallacy in a single government purchase creates infinite GDP?

Question

Suppose the government increase its spending by $100 million. And suppose we are talking about intermediate level macroeconomic class. Let the marginal propensity to spend be 0.5.

Now the total GDP should increase by 100/0.5=200 because the increased spending increases people's income and thus further increase the spending, so on and so forth. But wait, what if the marginal propensity to spend is 0! Then we have an infinite increase in GDP! OR if this sounds just wrong, let us consider where the unspent money get saved in banks. And banks will lend the money out which induce firm to invest more. So the investment will increase people's income, and hence the same effect as before. So in the end, we see that the increase in government spending will increase income by the same amount, and then increase spending again by the same amount (the saving also get invested, so it increases the GDP), and etc.

This is clearly not the case in the world we live in. What is the catch?

Answer 1

It seems that you misunderstood the concept of marginal propensity to consume. The marginal propensity is only valid for an infinitesimal increase in income. As soon as you move away from the original starting point where the marginal propensity was 1, there may be a different marginal propensity. The multiplier however assumes that the marginal propensity to consume remains the same at all income levels (since it is essentially an infinitely repeated application of the marginal propensity to consume).

The above problem with such analyses is why economists switched to models with so-called microfoundations in which the firm and consumer behavior is explicitly modeled via profit/utility maximization. In a model with fixed resources it would quickly become clear that a marginal propensity to consume of 1 for all income levels would violate market clearing conditions.

Answer 2

It appears that there exists a fascination with considering extreme values of mathematical formulas that are used in Economics (see for example questions about infinite elasticity etc) and wonder what will happen if these extreme values where actually observed, tending to forget the economics of the situation along the way.

So let's see: if marginal propensity to consume is constant (undergraduate macro) and equals unity, we have no saving whatsoever. So also, no investment. If we have no investment, the maximum productive capacity of the economy will remain fixed, or worse it will decrease due to depreciation of capital.

If the productive capacity remains fixed then, even if we assume that the increase in government spending happens at a time where there is no full employment of factors of production, the ultimate limit of the multiplier effect is the said maximum productive capacity plus the capital stock of the economy, assuming that we can sell it abroad and engage in a consumption orgy.

Write the macroeconommic identity (closed economy) as

$$F(K,L)+(1-\delta)K \equiv C_p + I_p +C_g$$

where $C_g$ is government consumption, not expenses. The left-hand side is all we got, after production takes place. Then with $I_p=0$ (since marginal propensity to consume is constant and unity) we have that, at most,

$$\max C_p = F(K,L)+(1-\delta)K < \infty$$

The "catch" is that a full specification of the "multiplier effect" would require a function with branches to deal with any extreme values that push the economic system to hit the wall of finiteness.

Answer 3

There's a steady state level of capital and prices of capital, and similarly a steady state for the labor supply. Banks at some point don't lend more out, or whatever lending they do, firms use to invest in capital to replace depreciation. I'd probably also hazard a guess that some investment is just repaying loans, so money velocity can't just grow forever. Those are the first things that come to my mind, but I'm sure there's a more direct answer to this.