Sometimes it's good to build really weird models. Because bits of the real world are a bit weird, even if the whole world isn't as totally weird as the model, and taking an extreme case can help us understand the effect of those weird bits. Plus it's fun.
I'm going to assume that all produced goods are strictly "non-rival".
Let me explain what "non-rival" means:
Apples are "rival", because if I get pleasure from consuming an apple, you can't get pleasure from consuming the same apple. The cost of producing 100 apples is the same whether 100 people eat 1 apple each, 10 people eat 10 each, or 1 person eats 100 apples.
Computer software is "non-rival", because if I use a program it doesn't prevent you from using another copy of the same program. The cost of producing 100 copies of the same program is (almost) the same as producing 1 copy of that program. Once 1 copy has been produced, the marginal cost of an additional copy is (almost) zero.
(A very good comment by rsj on my previous post about labour hoarding inspired this post. Especially when I realised I could recycle with one minor modification a model I wrote (pdf) when I was but a lad.)
I'm going to assume that the technology of producing all goods is identical to the technology of producing (strict, Samuelsonian) "public goods". But unlike true "public goods", which are both non-rival and "non-excludable", my goods are "excludable", meaning the producers can prevent people using them unless they pay the price the producer wants to charge.
So we are talking about an economy with extreme Increasing Returns to Scale. There are N different goods produced, and each good requires one unit of labour input, so N=L, where L is employment. But there are C copies of each good produced, so real GDP (total output or total expenditure or total income per year) is given by Y=C.N (the number of different programs times the number of copies per program). The real wage is W/P, which equals the Marginal Cost of producing one more variety. But the Marginal Cost of producing one extra copy of an existing variety is zero.
(It's easier to get your head around the model if you assume that a new software program lasts for only one year, then becomes obsolete and totally useless.)
Assume a very simple Aggregate Demand Curve P.Y=M. (Yes, yes, I know; it's ultra-monetarist, but it really doesn't matter.)
Assume perfectly flexible money wages and continuous full-employment, and an exogenously fixed supply of labour. So the number of different goods produced is also fixed exogenously. (Keynesians are now groaning, chortling with laughter, or sniggering uncontrollably; but just wait guys, and watch me!)
If prices are sticky in the short run, any change in Aggregate Demand, no matter how big a change it is, will cause an unlimited change in real output Y. Because there is no upper limit on the number of copies that can be made, even with finite resources. And firms will produce and sell as many copies as are demanded, at existing prices.
The Short Run Aggregate Supply curve is a very thick curve indeed, that covers any level of real GDP between zero and plus infinity.
And productivity (real GDP per worker employed) will rise in booms and fall in recessions. Just like in the real world. Because productivity is simply equal to the number of copies made of each variety.
What about the Long Run Aggregate Supply curve? What prices will firms set, in the Long Run when prices are perfectly flexible? That's the tricky bit.
Since the Marginal Cost of one extra copy is zero, profit-maximising firms will set prices where the Marginal Revenue of an extra copy is zero. Or where the individual firm's demand curve for copies of its variety, taking as given the prices set by all other firms and hence total output (or real income) Y (I'm assuming that there is a large number of small firms), is unit-elastic.
That's the first-order condition for a maximum.
The second-order condition for an individual firm (to ensure it is maximising and not minimising profits) is that the MR curve (drawn taking aggregate Y as given) is downward-sloping. Let's just assume it is downward-sloping.
But that's still not sufficient to define a unique and stable Long Run equilibrium at the macroeconomic level. To see this, consider the following:
Suppose we start in Long Run macroeconomic equilibrium. And then M increases, and the Aggregate Demand curve shifts right. In the short run, with sticky prices, all firms will sell more copies, and so real income will increase too. With Y higher, and each individual firm's demand curve and marginal revenue curve shifted to the right, will individual firms now want to increase their prices?
That depends. It depends on what happens to the elasticity of an individual firm's demand curve when Y increases and it shifts to the right (holding relative prices constant). It might become more or less elastic.
If it becomes less elastic in a boom, we get a unique and stable Long Run macroeconomic equilibrium, with a vertical LRAS curve. Because if AD increases, so Y increases in the short run (when prices are sticky), individual firms will find their demand curves become less elastic, so Marginal Revenue is now negative, and firms will want to raise their prices. A rising price level moves us back along the AD curve to the LRAS curve. So money is neutral in the long run.
But if it becomes more elastic in a boom, things get weird. Because if AD increases, so Y increases in the short run (when prices are sticky), individual firms will find their demand curves become more elastic, so Marginal Revenue is now positive, and firms will want to lower their prices. A falling price level moves us further along the AD curve to the right, which increases Y still more, and takes us further away from the LRAS curve. Even if Long Run macroeconomic equilibrium exists, it will be unstable. Update: The Short Run Phillips Curve slopes the wrong way. Prices start falling in a boom, and start increasing in a recession.