> So either a) Wikipedia is misrepresenting the issue, b) I'm making some
> glaring logical error, or c) I am the smartest person in the world. (c)
> seems rather less probable than the alternatives. If (a), what's a better
> summary? If (b), what embarrassing thing am I doing wrong?
I agree with others that the transformation problem is a swamp into which you don't want to get dragged. But you're basically asking 'why is there a profit rate at all?', or at least 'why is it exogenous?' The wikipedia article doesn't address this at all, but maybe it doesn't need to - because it's really a separate issue from the tranformation, which takes a profit rate for granted - it's exogenous, like you say.
I think the basic answer is that producing capital uses real resources and time, for which there are opportunity costs. So to get more arrows made, the entrepreneur has to get hunters to switch from hunting, but pay them at least the real wage they would have got hunting. So they will still consume as many beavers and deer as previously, but there will be fewer hunters 'producing' beavers and deer. So to make the story complete, you also need to know things like
- how many deer and/or beavers per day do people need to live on? - how many deer and how many beavers can hunters catch on average in a day without arrows, and with arrows? - how many arrows have already been accumulated?
The thing that stops people making infinite arrows (capital) and driving the profit rate down to zero is that accumulation of the arrows requires the diversion of workers away from hunting. In the example you get from Wikipedia, it seems there is no wage-labour so 'workers' and 'entrepreneurs' are the same people. In that case, there would be a tendency for any switching of effort to arrows to drive up the arrow-price of deer and beavers - and the limit point would be such (factoring in depreciation) that the rent of arrows would equal their (marginal) productivity in deer and beaver 'production', i.e. the extra amount of deer/beavers a single hunter can catch on average with an arrow over what they can catch without an arrow.
In the wage labour case - i.e., all hunting and arrow-making is done by employees of capitalists - the wage rate becomes an endogenous variable too. Arrow-making capitalists compete with hunting capitalists for workers, and if accumulation of arrows happens too fast, it tends to drive up beaver/deer-wages, cutting into the surplus capitalists end up with to devote to arrow accumulation. (This is the mechanism Marx emphasised.)
Mike Beggs