(a) Consider an open economy with a consumption matrix C...

(a) Consider an open economy with a consumption matrix $C$ whose column sums are less than $1,$ and let $x$ be the production vector that satisfies an outside demand d; that is, $(I-C)^{-1} \mathbf{d}=\mathbf{x} .$ Let $\mathbf{d}_{j}$ be the demand vector that is obtained by increasing the $j$ th entry of d by 1 and leaving the other entries fixed. Prove that the production vector $\mathbf{x}_{j}$ that meets this demand is $$\mathbf{x}_{j}=\mathbf{x}+j \text { th column vector of }(I-C)^{-1}$$ (b) In words, what is the economic significance of the $j$ th column vector of $(I-C)^{-1} ?\left[\text {Hint: } \text { Look at } \mathbf{x}_{j}-\mathbf{x} .\right]$

(a) Consider an open economy with a consumption matrix $C$ whose column sums are less than $1,$ and let $x$ be the production vector that satisfies an outside demand d; that is, $(I-C)^{-1} \mathbf{d}=\mathbf{x} .$ Let $\mathbf{d}_{j}$ be the demand vector that is obtained by increasing the $j$ th entry of d by 1 and leaving the other entries fixed. Prove that the production vector
$\mathbf{x}_{j}$ that meets this demand is $$\mathbf{x}_{j}=\mathbf{x}+j \text { th column vector of }(I-C)^{-1}$$
(b) In words, what is the economic significance of the $j$ th column vector of $(I-C)^{-1} ?\left[\text {Hint: } \text { Look at } \mathbf{x}_{j}-\mathbf{x} .\right]$

Key Concepts

Economic Multiplier Effect

The economic multiplier effect quantifies how an initial change in external demand in one sector leads to a magnified change in total production. Specifically, the j th column of the Leontief inverse provides a detailed account of the increase in production across all sectors when there is a unit increase in the external demand for sector j, reflecting the chain reaction of inter-sectoral demand.

Leontief Inverse

The Leontief inverse, often denoted by (I-C)?¹, encapsulates both direct and indirect effects of changes in final demand on total production. It mathematically unfolds the ripple effects through the inter-industry relationships and is used to determine the overall production levels needed to meet a given external demand.

Consumption Matrix

In input-output analysis, the consumption matrix represents the direct requirements of production. Each entry in the matrix shows the proportion of output from one sector necessary as an input for another sector's production. The condition that column sums are less than 1 is crucial as it reflects the net addition of value and ensures the system is economically viable.

Input-Output Analysis

This is a framework for modeling the interdependencies between different sectors within an economy. It examines how the output from one industry is an input to another, thereby capturing both direct and indirect effects of a change in one sector on the entire economy.

Transcript

Please give Ace some feedback