I am trying to understand what drives differences between average individual consumption growth and average aggregate consumption growth.
Let's assume there are $N$ agents and $T$ time periods and $K$ economies. There is a matrix $C_k$ which stores the consumption of each agent $n$ and each time period $t$ for each economy $k$. This is a $[T \times N]$ matrix.
\begin{equation} C_k = \begin{bmatrix} c_{k,1,1} & c_{k,2,1} & ... & c_{k,N,1} \\ c_{k,1,2} & c_{k,2,2} & ... & c_{k,N,2} \\ ... & ... & ... & ... \\ c_{k,1,T} & c_{k,2,T} & ... & c_{k,N,T} \\ \end{bmatrix} \end{equation}
So $c_{k,i,t}$ is the consumption of agent $i$ at time $t$ in economy $k$.
Now I can compute each agent's individual consumption growth $\Delta c$:
\begin{equation} \Delta C_k = \begin{bmatrix} \Delta c_{k,1,2} & \Delta c_{k,2,2} & ... & \Delta c_{k,N,2} \\ \Delta c_{k,1,3} & \Delta c_{k,2,3} & ... & \Delta c_{k,N,3} \\ ... & ... & ... & ... \\ \Delta c_{k,1,T} & \Delta c_{k,2,T} & ... & \Delta c_{k,N,T} \\ \end{bmatrix} \end{equation}
Where $\Delta c_{k,n,t} \equiv ln(\frac{c_{k,n,t}}{c_{k,n,t-1}})$.
So the average individual consumption growth across all economies, time periods and agents is:
\begin{equation} Average IndividualConsumptionGrowth=\sum^K_{k=1} \sum^N_{n=1} \sum^T_{t=2} \frac{\Delta c_{k,n,t}}{K \times N \times (T-1)} \end{equation}
Now I also want average aggregate consumption growth. First we need a vector of aggregate consumption which is just the sum of individual consumptions.
\begin{equation} C^{agg}_k = \begin{bmatrix} \sum_{n=1}^N c_{k,n,1} \\ \sum_{n=1}^N c_{k,n,2} \\ ... \\ \sum_{n=1}^N c_{k,n,T} \\ \end{bmatrix} \end{equation}
Which gives aggregate consumption growth as:
\begin{equation} \Delta C^{agg}_k = \begin{bmatrix} \Delta \sum_{n=1}^N c_{k,n,2} \\ \Delta \sum_{n=1}^N c_{k,n,3} \\ ... \\ \Delta \sum_{n=1}^N c_{k,n,T} \\ \end{bmatrix} \end{equation}
Where $\Delta \sum_{n=1}^N c_{k,n,t} \equiv ln(\frac{\sum_{n=1}^N c_{k,n,t} }{\sum_{n=1}^N c_{k,n,t-1} })$.
Then average aggregate consumption growth follows to be:
\begin{equation} Average AggregateConsumptionGrowth=\sum^K_{k=1} \sum^T_{t=2} \frac{\Delta \sum_{n=1}^N c_{k,n,t}}{K \times (T-1)} \end{equation}
So the question is what are the properties of the relation between:
$\sum^K_{k=1} \sum^T_{t=2} \frac{\Delta \sum_{n=1}^N c_{k,n,t}}{K \times (T-1)}$ and $\sum^K_{k=1} \sum^N_{n=1} \sum^T_{t=2} \frac{\Delta c_{k,n,t}}{K \times N \times (T-1)}$
Can one term be negative and the other positive? What drives the differences between the two terms?
