The Leontief production equation \({\bf{x}} = C{\bf{x}} + {\bf{d}}\), is usually accompanied by a dual price equation,

\({\bf{p}} = {C^T}{\bf{p}} + {\bf{v}}\)

Where \({\bf{p}}\) is a price vector whose entries list the price per unit for each sector’s output, and \({\bf{v}}\) is a value added vector whose entries list the value added per unit of output. (Value added includes wages, profit, depreciation, etc.). An important fact in economics is that the gross domestic product (GDP) can be expressed in two ways:

{gross domestic product} \( = {{\bf{p}}^T}{\bf{d}} = {{\bf{v}}^T}{\bf{x}}\)

Verify the second equality. [Hint: Compute \({{\bf{p}}^T}{\bf{x}}\)in two ways.]

The given equation is \({\bf{p}} = {C^T}{\bf{p}} + {\bf{v}}\),

\(\begin{array}{c}{{\bf{p}}^T} = {\left( {{C^T}{\bf{p}} + {\bf{v}}} \right)^T}\\ = {\left( {{C^T}{\bf{p}}} \right)^T} + {{\bf{v}}^T}\\ = {{\bf{p}}^T}C + {{\bf{v}}^T}\end{array}\)

Multiply the above equation by \({\bf{x}}\) on both sides.

\(\begin{array}{c}{{\bf{p}}^T}{\bf{x}} = \left( {{{\bf{p}}^T}C + {{\bf{v}}^T}} \right){\bf{x}}\\ = {{\bf{p}}^T}C{\bf{x}} + {{\bf{v}}^T}{\bf{x}}\end{array}\)

Multiply the equation \({{\bf{p}}^T} = {{\bf{p}}^T}C + {{\bf{v}}^T}\) by \({\bf{x}}\) on both sides.

\(\begin{array}{c}{{\bf{p}}^T}{\bf{x}} = \left( {{{\bf{p}}^T}C + {{\bf{v}}^T}} \right){\bf{x}}\\ = {{\bf{p}}^T}C{\bf{x}} + {{\bf{v}}^T}{\bf{x}}\end{array}\)

Find the value of \({{\bf{p}}^T}{\bf{x}}\) using the equation \({\bf{x}} = C{\bf{x}} + {\bf{d}}\).

\(\begin{array}{c}{{\bf{p}}^T}{\bf{x}} = {{\bf{p}}^T}\left( {C{\bf{x}} + {\bf{d}}} \right)\\ = {{\bf{p}}^T}C{\bf{x}} + {{\bf{p}}^T}{\bf{d}}\end{array}\)

\(\begin{array}{c}{{\bf{p}}^T}C{\bf{x}} + {{\bf{v}}^T}{\bf{x}} = {{\bf{p}}^T}C{\bf{x}} + {{\bf{p}}^T}{\bf{d}}\\{{\bf{v}}^T}{\bf{x}} = {{\bf{p}}^T}{\bf{d}}\end{array}\)

So, the equation\({{\bf{v}}^T}{\bf{x}} = {{\bf{p}}^T}{\bf{d}}\)is true.