This is mainly a brain-teaser. But I hope you find it useful, as well as fun.
1. Assume initially there is no government. So G=T=0, and Y=C+I. (You can add in net exports too if you want, or delete I if you want to make the model simpler still.)
2. Assume MV=P(C+I), where V is fixed and C+I is output of consumption plus investment goods that are bought and sold for money.
3. Assume a production function Y=L where L is employment.
4. Assume a perfectly inelastic labour supply function Ls=8. People want to work 8 hours per day, regardless.
5. The economy starts in full-employment equilibrium with MV/P=C+I=Y=L=8.
6. Then something causes M to halve, or V to halve. P is fixed, and so Y and L both halve to 4. There is involuntary unemployment.
7. Then a government comes along and says "Right you lot, instead of spending 4 hours a day sitting on your arses doing nothing you are going to spend 4 hours a day building pyramids for us. We won't pay you any money for building pyramids, but we will punish you if you don't."
8. M, V, and P stay the same, so C+I stays the same at 4. Remember that C+I is defined as output that is bought and sold for money. The pyramids, and the labour used to build them, are not bought and sold for money.
9. Statistics Canada values the pyramids at input cost, and it took 4 hours of labour per worker per day to build the pyramids, and 1 unit of labour produces 1 unit of output, so G rises by 4. Statistics Canada also notes that taxes must have increased by 4, since the government did not issue any bonds or print any money to pay for the pyramids.
10. Economists estimate that the balanced budget multiplier, defined as the change in C+I+G divided by the change in G that brought it about, assuming the change in G is equal to the change in taxes, is exactly equal to one.
My model of the government expenditure multiplier is much simpler than Mike Woodford's (pdf) very sophisticated model. My model is also very monetarist, and his is very New Keynesian. But we get exactly the same answer.
So I'm going to leave you three puzzles:
1. How come two extremely different models come up with exactly the same answer? Are the two models really the same underneath, even though they look very different?
2. If my model is wrong because taxes and government spending are in fact paid in money, does that make Mike Woodford's model wrong too? Where does it say in Mike Woodford's model that taxes and government spending are paid in money?
3. My model has involuntary unemployment. In Mike Woodford's model, employment is always equal to the quantity of hours workers want to work at the market wage. Which model is more Keynesian? (OK, that was slightly mischievous!)
If you like, forget Mike Woodford's model, and just think about the very simple Old Keynesian model, where the balanced budget multiplier is also equal to one.
And have a look at Lars Christensen's post too. (I read Lars' post just after writing the above, and see we are heading in a similar direction.)