If all labor was physical and I invented a pill that doubled everybodies strength, what would actually happen to both prices, unemployment and wages?
This 'toy example' does not have enough assumptions to give clear answer. Assuming that output is affected by the strength, labor is the only factor of production and we assume market is competitive then labor demand can be derived from profit maximization of competitive firm, e.g.
$$\max pq(L) -c(L); \quad q(L) = AL^\beta, c(L)=wL$$
Where $p$ is price, $q$ is production which depends on labor $L$ input that can be augmented by labor enhancing technology $A$, and $c$ is cost function that is simply equal to wage $w$ times labor used. We get that:
$$ L^* = \left(\frac{pA(1-\beta)}{w}\right)^\frac{1}{1-\beta}$$
So increasing labor productivity through the 'strength enhancement' ($A$) will increase labor demand and unemployment goes down. This is because it allows firm to sell more products.
What happens to wages? We can rearrange the result for $w$ to get:
$$ w= pA(1-\beta)L^{(\beta-1)} \quad or \quad w/p= A(1-\beta)L^{(\beta-1)}$$
So increase in productivity $A$ raises both nominal $w$ and real $w/p$ wages.
What happens to prices? We can solve for $p$:
$$ p= \frac{w}{A(1-\beta)L^{(\beta-1)}}$$
Increase in productivity of labor lowers the prices consumers pay.
Of course, different assumptions would change the result, but generally speaking for large set of broad assumptions:
- increase in labor productivity leads to higher real wages (although in richer models this depends also on other factors such as bargaining power/union strength etc). You can find some models where wages would fall but in most they would raise.
- prices will drop. This result is again quite robust, although how much they drop depends on how competitive market is.
- unemployment is very sensitive to assumptions being made. Depending how we model production function, and whether technology makes labor or capital more productive, or depending on rate at which labor can be substituted for capital you can get widely different results regarding effect on employment.
What happens to the unemployment rate? In reality, labor is not the only cost factor, and so there is still a limit to how cheap I can make a good. Wouldn't that just mean I need half the people to produce the same amount of product (I can not increase the amount of product I produce since other cost factors are still limited), and would let half the people go?
No that is too simplistic.
Macroeconomically production is equivalent to income as economically speaking national income is not money but goods and services a nation can produce and thus enjoy in a given period of time (see discussion in any macro textbook for example Blanchard et al Macroeconomics).
When labor becomes more productive more goods and services can be produced, which increases national income, which increases further consumption spending. Hence, when people get more productive instead of firing the workers firm can produce more and sell more because consumers will consume more goods and services.
Of course, as mentioned above this result is contingent on number of factors such whether technology actually makes labor more productive, or whether technology actually makes capital more productive and allows capital to be substituted for labor (e.g. some robots or some form of AI that directly substitutes for labor). If economy has production function with high degree of substitution between capital and labor, and it is actually capital that is getting more productive then you will generally find that technology leads to more unemployment.
If you want some literature on this topic just put technological unemployment or capital-labor substitution into google scholar, there is quite wide literature on the issue.
Since technology removes jobs (directly, as in: Trucks now drive without a driver and indirectly like here: I now need less steelworkers for the same amount of product), is there any way to predict how many new jobs a new technology creates and in what sector?
You can use models to predict/forecast anything (of course accuracy of such predictions/forecasts is another matter), but that is too broad of a topic to cover in any detail here. One way how you can forecast it would be to estimate industry level production function, then depending on the estimates you can predict how change in productivity of labor would affect labor demand in the particular industry.
For some literature that sorta tries to look at what industries will be affected by technological unemployment you can check Kim et al 2017 and sources cited therein.
