Multiplier-accelerator model: Difference between revisions

Revision as of 23:59, 21 October 2015

Revision as of 00:04, 22 October 2015

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:<math>I_{t} = \beta \mid C_{t} - C_{t-1} \mid </math>

:<math>g_{t} = 1</math>

where <math>Y_{t}</math> is national income, <math>g_{t}</math> is government expenditure, <math>C_{t}</math> is consumption expenditure, <math>I_{t}</math> is induced private investment, and the subscript <math>t</math> is time. Here we can rearrange these equations and rewrite them as a second -order linear [[difference equation]]:<ref name="Mullineux 1984" /><ref name="Goldberg1958">{{cite book |last=Goldberg |first=Samuel |title=Introduction to Difference Equations |___location=New York |publisher=John Wiley & Sons |year=1958 |pages=153–56 }}</ref><ref>{{cite book |first=Giancarlo |last=Gandolfo |authorlink=Giancarlo Gandolfo |title=Economic Dynamics |___location=Berlin |publisher=Springer |edition=Third |year=1996 |chapter=Second-order Difference Equations in Economic Models |pages=71–81 |chapterurl=https://books.google.com/books?id=ZMwXi67nhHQC&pg=PA71 }}</ref>

:<math>Y_{t} = 1 + \alpha (1+ \beta)Y_{t-1} - \alpha \beta Y_{t-2}</math>

Samuelson demonstrated that there are several kinds of solution path for national income to be derived from this second order linear difference equation.<ref name="Mullineux 1984" /><ref name="Goldberg1958" /> This solution path changes its form, depending on the values of the roots of the equation or the relationships between the parameter <math>\alpha</math> and <math>\beta</math>.<ref name="Mullineux 1984" /><ref name="Goldberg1958" />