I'm reading an economics paper in which technical knowledge at time t ($A_t$) is the function of past production ($y_{i,t-1}$) for each individual and the learning done by one individual affects the level of technology of all other agents in the economy.
So I've got
$A_t = \int y_{i,t-1} di $
According to the paper, this integral equals $ y_{t-1}$
What does it mean to take the integral with respect to $i$, (i.e. the index of individuals)? Is it a sort of sum? And is it solved using the usual integration rules? Thanks.
The authors appear to be looking at the average production level at time $t-1$ by integrating over all individuals $i$ in the economy. Macroeconomists often take the average production instead of the total production, which is the same except the average divides by a constant. They apparently use a $U[0,1]$ density, or the $di$ is shorthand for $f(i) di$ (i.e., integrating over all individuals, weighting by their density $f(i)$).
If there were finitely many individuals, they would be writing something like $$A_t=N^{-1}\sum_{i=1}^N y_{i,t1}.$$