The point is to keep trying until you get it right. I am sorry to say that I got it wrong last time, so I’m taking another shot at it.
Let’s consider, as does Simon Wren-Lewis a two-period model. The first period is in underemployment equilibrium. Let’s say that consumption in period 1 is given by the equation
C0 = 100 + bY0,
where b represents the marginal propensity to consume out of income.
Let’s say that investment is a fixed amount:
I0 = 100.
The expenditure (aggregate demand) equation is thus
E0 = 200 + bY0.
The equilibrium is determined by applying the equilibrium condition E0 = Y0, which gives us
Y0 = 200/(1-b).
Now the case that I posited in my previous post involved b = 0, reflecting income smoothing. This is tricky, because we have to make an assumption about what households expect their income to be in the next period, which can be assumed to be long relative to the initial period, though for simplicity I’m going to let the two periods be equal in length. If households expect income in the next period to reflect full employment, presumably they would try to increase their consumption now, spending more and increasing equilibrium income now, so there is an inherent inconsistency in the model which needs to be resolved, but I am not going to worry about that either. Let’s just take the model at face value.
In this equilibrium, note that consumption, C0, is 100, investment, I0, is 100, and saving, S0, is also 100.
What happens if the government immediately tries to intervene to raise income by increasing government spending, G0, from 0 to 100, and imposes taxes, T0, of 100 to finance its spending? The increased spending is only for this period and the taxation is only for this period, not the next one; in period 1, government spending and taxation go back to zero. What this does is to cause the consumption function to be revised as households choose a uniform level of consumption to be maintained for both periods, reflecting the liability to pay taxes this period, but no obligation to pay taxes next period.
Expecting income next period of 200, households would have chosen to consume 100 this period and 100 next period. But with a tax liability of 100 this period, households will choose, instead of consuming zero this period and 100 next period, to consume 50 this period and 50 next period. They have to borrow 50 this period to be able to pay their tax liability in order to have 50 left over for consumption. Next period, they will have to repay the loan of 50, and will have only 50 left over for consumption (income remaining at 200 with consumption equal to 50 and investment equal to 50, the loan repayment of 50 corresponding, it seems to me, to exports to shipped to foreigners). So the new consumption equation is
C’0 = 50 + bY0, where b is again equal to 0.
Now adding government spending and taxes, we have G = 100 and T = 100, so our new expenditure equation becomes
E’0 = 250 + b(Y’0 – 100).
But since b = 0, this reduces to
E’0 = 250 = Y’0.
We still have I0 = S0 = 100. Since b = MPC = 0, (1-b) = MPS = 1. The increase in income from 200 to 250 is just enough to generate another 50 in savings to offset the 50 in borrowing required to keep consumption level at 50 in period 0 and period 1.
The increase in government spending and taxes of 100 in period 0 raises the period-0 equilibrium (as compared with the case with no government spending and taxes) is 50, so the multiplier is .5.
Of course, this is not a full-equilibrium solution. A full equilibrium should have Y1 also equal to 250 instead of 200, which means that consumption could have been increased by 25 in both periods, but I haven’t worked that solution out yet.
The reason why in this post I arrive at a result different from the result in my previous post is that I made a simple flunk-the-quiz mistake in the previous post, reducing the expenditure curve by 100 to reflect the reduction in disposable income from taxes as if it were a permanent reduction in disposable income rather than a one-period reduction in disposable income. So instead of assuming the MPC was 0 as I wanted to do, I was assuming, for purposes of the effect of taxes on consumption, an MPC of 1. Yikes! My assertion that everything depended on a positive MPC was entirely wrong. In a simple Keynesian model, you get a balanced-budget multiplier of 1 provided that the MPC is less than 1. That was a pretty bad blunder on my part, and I apologize. Scott, himself, seemed to perceive that something was amiss in a comment on the previous post, so I hope that we are now converging toward a solution.
Again my apologies for hastily posting my previous post without checking my work more carefully. I had better get some rest now.