If C, P and R are column vectors and A and B are the matrices of the input coefficients then the matrix version of the above equations is

C = A^TP + B^TR

where A^T and B^T represent the transpose of the matrices A and B, respectively. The transpose of a matrix A is the matrix in which the rows of A are written as the columns.

Having introduced the matrix B and the price vector R as the inputs requirements and prices of imported goods their nature can be made more general. They represent the input requirement and prices for all goods whose prices are not influenced by domestic prices. For example, if wage rates are not set by supply and demand but instead by government fiat then labor inputs and wages rates would be included in B and R. Of course, if the government set wage rates based upon the cost of living then labor inputs and wage rates would just be another element of the A matrix and price vector P.

If the prices are equal to the cost of production; i.e., P=C, then

P = A^TP + B^TR

This system can be solved as follows:

IP − A^TP = (I−A^T)P = B^TR
P = (I−A^T)^-1B^TR

Thus the matrix (I−A^T)^-1B^T gives the impact on domestic prices of a change in the price of imported material requirements. This gives the complete answer to the question of the impact of import prices on domestic prices, assuming linearity. This could be characterized as the case of perfect competition.

Oligopolistic Pricing

Suppose the demand function for a market is linear so the inverse demand has the form

p = p_max − hq

where q is quantity and p_max is the intercept of the demand function with the price axis. The intercept of the demand function with the horizontal axis is p_max/h. This is the quantity which would be consumed if the commodity were free, say q_max. The quantity q_max would not depend upon consumer incomes but p_max would. This means that h the slope of the inverse demand function is equal to p_max/q_max, and thus is dependent upon consumer incomes.

A monopolist with a constant marginal cost of c when confronted with the above inverse demand function will set a price equal to

p_mon = ½p_max + ½c

The above formula can be generalized to the case of oligopoly. In a market characterized by conjectural variation models of oligopoly if the inverse demand function is of the above form then the oligopoly price is of the form

p_olig = (1/(n+1))p_max + (n/(n+1))c

where c is the marginal cost and n is a factor which is just the number of firms in the market if the firms are Cournot-type oligopolists. More general n is the sum of the competitive weights of the firms in the market. The competitive weight of a Cournot-type oligopolist who take the other firms' output as given have a weight of unity. If some of the firms are leaders which take into account the reaction they expect from the other firms then n is larger than unity. Von Stakelberg oligopolists have a weight of two.

The case of monopoly is just the case in which n=1. As the value of n increases the role of p_max in price determination goes to zero and n/(n+1) goes to 1.0 and price is determined by cost alone.

For the determination of oligopoly prices let N and M be diagonal matrices with (1/(n_i+1)) and n_i/(n_i+1) on the diagonals, respectively. Then the matrix equation for price determination is

P = NP_max + M(A^TP + B^TR)

This system can be solved as follows

IP − MA^TP = NP_max + MB^TR
or, equivalently
(I − MA^T)P = NP_max + MB^TR
and finally
P = (I−MA^T)^-1NP_max + (I−MA^T)^-1MB^TR

The matrix (I−MA^T)^-1MB^T gives the complete picture of the impact of imported material prices (or any other autonomous prices) on domestic prices taking into account the degrees of oligopoly in each industry.

When people have higher income then the values in P_max increase and prices go up simply because people will pay more. The prices now charged for foo-foo coffees in coffee shops and elegant ice creams have no other explanation. This is a neglected source of inflation.