SNA: Constant Price Estimates

The current value of a product can be expressed as the market price per unit times the number of units. If v is value, p is price and q is quantity, then; v_i,t = p_i,t*q_i,t

The value of a product can, instead of being expressed in the market price of the current period be expressed in the market price of some previous (base) period; kv_i,t = p_i,0*q_i,t

Measuring transactions for a product at constant prices means to construct a time series where all transactions in that product are expressed in the price of a selected base year. This gives a series;
kv_i,0 = p_i,0*q_i,0, kv_i,1 = p_i,0*q_i,1, kv_i,2 = p_i,0*q_i,2, … , kv_i,t = p_i,0*q_i,t

It follows from the relationship above that the current price value of the product can be expressed as the increase in the price of the product from the base period to the current period times the constant price value of the product;
v_i,t = p_i,t/p_i,0*kv_i,t = p_i,t/p_i,0*p_i,0*q_i,t

A similar relationship can be established on an aggregated level;
V_t = S_iv_i,t = S_ip_i,t*q_i,t = [S_ip_i,t*q_i,t / S_ip_i,0*q_i,t ] * S_ip_i,0*q_i,t = PP_0,t * S_ip_i,0*q_i,t

The value at current price equals the price index, a Paasche price index, times the constant price value. Vice versa is the constant price value then estimated as the current price value divided by the price index. The Paasche price index is one out of several alternative aggregated price index measures.

Constant price measures are monetary measures of some aggregates where each individual quantity is valued at its own price in an earlier period, and these prices are kept constant. Since the prices on each item is kept fixed, or constant , the period to period changes in constant prices reflect the changes in the quantities (and qualities) of the different products. In this sense, constant price measures are “volume” measures.

The change from the base period to another at constant prices constitutes a volume index. The Laspeyres volume index is derived as;LQ_0,t = S_ip_i,0*q_i,t / S_ip_i,0*q_i,0

From this, a relationship between value, price and volume indices can be established;
V_t / V₀ = S_ip_i,t*q_i,t / S_ip_i,0*q_i,0 = S_ip_i,0*q_i,t / S_ip_i,0*q_i,0 * S_ip_i,t*q_i,t / S_ip_i,0*q_i,t = LQ_0,t * PP_0,t

SNA recommends the use of Fisher price and volume indices, Fisher's Ideal Indices.

However, most countries use the combination of Laspeyres volume indices and Paasche price indices rather than the Fisher indices. The Fisher indices are 'ideal' compared to Laspeyres and Paashe from a theoretical point of view, but (i) Fisher requires that both Laspeyres and Paasche (volume and price) indices are calculate, and (ii) only if the Laspeyres formula underlies the constant price estimates (e.g., if a Paasche price indices are used to deflate current price values) will the constant price estimates be additive.