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The Leontief production equation, $\mathbf{x}=C \mathbf{x}+\mathbf{d},$ is usually accompanied by a dual price equation,$\mathbf{p}=C^{T} \mathbf{p}+\mathbf{v}$where $\mathbf{p}$ is a price vector whose entries list the price per unit for each sector's output, and $\mathbf{v}$ is a value added vector whose entries list the value added per unit of output. Value addedincludes wages, profit, depreciation, etc. ) An important fact in economics is that the gross domestic product (GDP) can be expressed in two ways:$\{\text { gross domestic product }\}=\mathbf{p}^{T} \mathbf{d}=\mathbf{v}^{T} \mathbf{x}$Verify the second equality. [Hint: Compute $\mathbf{p}^{T} \mathbf{x}$ in two ways.

$\mathbf{p}^{\mathrm{T}} \mathbf{d}=\mathbf{v}^{\mathrm{T}} \mathbf{x}$

Algebra

Chapter 2

Matrix Algebra

Section 6

The Leontief Input–Output Model

Introduction to Matrices

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In this example, we're going to be considering the Leon Tiff production equation, which is given here, as well as the dual price equation where P is equal to see transpose p plus v. Here he is a price vector and he is the value added vector. And we also have an equation that tells us that the GDP equation is GDP equals p transposed times D, which is also equal to be transposed times D. So our goal here is to show that these two quantities are in fact equal as the GDP equation tells us it should be. Let's start off with the dual price equation that P is equal to see transposed p plus V. Now I'm going to look at P transpose coming from the left hand side of this equation we have P transposed is going to be the transpose of this entire group. So we'll have see transposed p plus v transposed and we get next the following first note that when we take the transpose of this product, we have p transpose times see transposed transposed, which is C So we obtain the transpose is in the reverse order of the product. Then add be transposed together. So this is what P transpose equals. If we were to take, say, P transpose times a vector X, then this would result in p transposed times the production matrix c times the production vector X plus the value that it vector transposed times the production vector x again. So this is what P Transpose X would be equal to. But we also know that X is equal to this quantity. So let's take another look at the equation that we had from here. We have P transpose equals p transposed time See plus V transposed. If we take p transposed times the vector X again and using the fact that X is equal to see X plus d, we'll obtain that this is equal to p transpose times c x plus deep. Then this tells us altogether p transposed Times X is equal to If we distribute this vector on the inside p transposed times c x plus be transposed times deep. So now we have two expressions for P transpose x, and they both include this term here p transposed time. See Time's X. That means if we set these two quantities equal to each other. We're able to conclude immediately that V transpose X coming from the left hand side is equal to p transpose times d coming from the right hand side and this verifies the equation above.

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