More about Partitions

More about Partitions

When analyzing firm sales, it's often useful to group firms into a family of non-overlapping subsets called parts (similar to sectors). Both aggregate sales and their variance are defined in an analogous way for firms and sectors, with the only difference being the index used.

$$X_t = \sum_k S_{kt} = \sum_p S_{pt}$$

Here, the indices $k$ represent firms and $p$ represent parts. Partitions can be determined by various criteria, such as industry sectors, geographical regions, or even random allocation.

Footnote 1: Partitions can also be defined on the sets of buyer firms as well as seller firms, allowing for a network approach to cross-accounting of sales. Using indices $p$ and $r$ for parts, total sales from exchanges between parts and firms can be expressed as:

$$X = \sum\limits_{p,\ r} s_{pr} = \sum\limits_{p,\ r} \sum\limits_{k,\ l \in p,\ r} s_{kl}$$

All sales from firm $k$ are associated with a firm $l$ on the other end.

$$X = \sum\limits_{k} S_{k} = \sum\limits_{k,\ l} S_{kl}$$

Here, $S$ represents the value of sales, and $k$ and $l$ represent a pair of firms.

Footnote 2: At a micro level, sales between a pair of firms during a given time period can be disaggregated into transactions $i$, composed of sales of items $j$ in quantities $q_j$ at prices $p_j$. The value of a transaction $s_j$ is in units of currency, derived from $s = p.q$. Aggregation is thus in units of currency. Prices can be assigned to each item $j$ when observed, without assuming they belong to a product, are time-dependent, or are consistent across different agents exchanging the same product.

$$S_{k} = \sum\limits_i t_i = \sum\limits_i \sum\limits_{j \in t_i} {p_j q_j}$$

In this work, $S_{k}$ is taken as given, focusing on aggregation up to the national level. Firm-level sales and atomistic items exchanged relate exactly in an ordinary sum.

$$X = \sum\limits_{k} S_{k} = \sum\limits_{j} {p_j q_j}$$

Types of Partitions

Two types of partitions are particularly useful due to their formal properties:

• Equal Weight Partitions: These are partitions where the value held by any $P$ parts is the same, ideally $\bar S_p = \bar X/P$. While perfect precision may not be achievable, firms can often be separated into parts with weights close to $\bar X/P$. Randomized assignments to parts can be referred to as random partitions.

• Quantile Partitions: Also known as size-sorted equal weight partitions, these involve sorting firms by size before dividing them into $Q$ groups. Each partition's total sales are near $\bar S_q = \bar X/Q$. With a sufficient number of parts (e.g., $Q = 10, 20$), firms in most parts are close to a mean size ($\bar s_q$).

For quantile partitions, we expect $\bar S_q = \bar X/Q = \bar s_q n_q$, where $n_q$ is the number of firms in quantile $q$. The mean size $\bar s_q$ increases monotonically across successive parts $q$, necessitating a decrease in quantile population $n_q$ to maintain the quantile value near $\bar S_q = \bar X/Q$. Quantile partitions thus offer defined agent and population sizes, making them highly useful for analyzing micro moments as functions of agent size and quantile $q$.

Note: Grouping firms can reduce the impact of multiplicative shocks on terms that sum to $X_t$. Linear relationships may apply to sectoral accounts even if they don't at the firm level. The problem of explaining parts' sales in terms of firm sales will be addressed in a later section.