Why Average Individual and Aggregate Consumption Growth Differ
Summary
The document defines two ways to measure average consumption growth across economies and time: averaging each agent’s log consumption growth, and first summing agents’ consumption within each economy and period, then calculating log growth in that aggregate. It asks how these averages relate and whether one can be positive when the other is negative, in the context of consumption-based asset pricing.
The key distinction is that the logarithm of a sum’s growth is not generally the same as the average of individual log growth rates. The relationship can depend on how consumption levels and growth rates vary across agents and over time, including changes in their relative contributions to the aggregate. The document poses this as a question and supplies definitions, but no derivation, empirical evidence, or conditions for a sign reversal. Its setup therefore frames an aggregation issue rather than resolving it; additional assumptions about weighting, population, and observations would be needed for a specific conclusion.
Key ideas
- Individual growth is defined as each agent’s log change in consumption.
- Aggregate growth is calculated from the sum of agents’ consumption at each time.
- The two averages use different orders of aggregation and logarithmic transformation.
- Differences may depend on consumption levels and how agent growth rates vary over time.
- The document asks about sign reversals but does not derive conditions or provide empirical findings.
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Full text
# Average individual consumption growth vs average aggregate consumption growth
# Average individual consumption growth vs average aggregate consumption growth
Consumption growth is an essential thing in most asset pricing models and usually the Euler equation defines the return of an asset as a covariance between consumption frowth and the cash-flows of that asset.
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.