On PSST, my former thesis adviser writes via snail mail,
Obviously I don't have to be told that there are very many outputs and very many distinguishable inputs. A question arises whether aggregation of some degree is a last resort, a confession of failure, or a positively good thing, because useful approximation is a good thing.
Instinctively, I like aggregating total hours worked, and I do not like the idea of an aggregate production function. Let me see if I can justify that instinct.
I think that economic activity consists of taking advantage of specialization and trade. When you work in the market, you are doing a relatively narrow, specific job in order to be able to consume a remarkable variety of goods and services that would be much harder for you to provide for yourself. So, the more hours you spend doing market work, and the fewer hours you spend trying to produce your own consumption goods, the better, with a caveat.
The caveat is planned leisure. Hours worked could decline over time because people intentionally take more leisure. So it is good when people either have market work or take planned leisure. It is bad when they engage in home production and when they have unplanned leisure (aka involuntary unemployment). For aggregate hours worked to mean what I want it to mean, you have to adjust for trends in planned leisure.
A related caveat is housework as leisure. Some people like to cook, or garden, or build furniture, even though that is not their comparative advantage in the market. I would say just count the time that people spend on those activities as planned leisure.
Once you have a handle on how much people are working in the market, it is tempting to try to aggregate the value of what they produce. Hence nominal GDP. And real GDP, which requires more assumptions but is a more powerful measure of value.
I think where macro starts to go off the rails is when folks try to talk about potential real GDP. What they assume is that if more people were working, these additional workers would produce real output at the same rate as the people who are working now. That is the aggregate production function that I do not buy into. I think that if people are not working now, it is because at the moment the economy is not organized in a way that enables them to produce anything of value. Assigning a value to what they could be producing if they were producing something of value is like the old joke "If we had some ham, we could have ham and eggs for breakfast--if we had any eggs." As of now, they have Zero Marginal Product (to use Tyler Cowen's phrase), or more accurately, a marginal product that is below the minimum needed to justify hiring them.
It is the task of entrepreneurs to organize the economy so that people produce stuff that has value. Sometimes, entrepreneurs are not quite up to that task. Then you get unemployment. There is an incentive for entrepreneurs to try to figure out ways to create patterns of sustainable specialization and trade that utilize workers who are currently unemployed. However, it is not as easy as you would imagine if you only think in terms of an aggregate production function, where everybody works in the GDP factory.
In short, total employment is a useful aggregate. Total output is a useful aggregate. Potential GDP is not a useful aggregate, because it makes the unjustifiable assumption that people who are not working now could produce something of value if they were working. That assumes away the problem, which is to re-organize production and trade so that more people can produce something of value.