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A Model of Growth, Debt and Taxes

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A Model of Growth, Debt and Taxes 

 

The following was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com 

and is based upon the work by Shone (2003) Economic Dynamics: Phase Diagrams and their Economics Application 

 

 

 

 

 

We assume that debt in the economy evolves according to
 

 

 

 

 

where:
 

= The amount of Debt at time t+1 

 

=  The amount of Debt at time t 

 

The amount of Government Spending at time t

= The amount of Transfer Payments at time t

=  The amount of Taxes at time t

The amount of Printed Money at time t (Seniorage)

Economic Growth Rate at time t 

 

Interest Rate at time t 

 

The positive effect economic growth has on debt at time t 

 

The cost of debt at time t 

 

 

 

 

The above equation can be written as:

 

 

 

D[`+`(t, 1)] = `+`(D[t], GS[t], TP[t], `-`(T[t]), `-`(PM[t]), `-`(`*`(`+`(g[t], `-`(r[t])), `*`(D[t])))) (1)
 

 

 

 

We now note that we can express the above equation as a difference equation if we subtract on both sides   

 

 

 

`+`(D[`+`(t, 1)], `-`(D[t])) = `+`(GS[t], TP[t], `-`(T[t]), `-`(PM[t]), `-`(`*`(`+`(g[t], `-`(r[t])), `*`(D[t])))) (2)
 

 

 


We now note that when the time increment in form of the difference between and is small we get: 

 

 

 

 

 

 

 

This means that we can write our difference equation as a differential equation as follows: 


 

 

diff(D(t), t) = `+`(GS[t], TP[t], `-`(T[t]), `-`(PM[t]), `-`(`*`(`+`(g[t], `-`(r[t])), `*`(D[t])))) (3)
 

 


at the fixed point which means that the above equation becomes 

 



 

0 = `+`(GS[t], TP[t], `-`(T[t]), `-`(PM[t]), `-`(`*`(`+`(g[t], `-`(r[t])), `*`(D[t])))) (4)
 


if we solve for we get:

 

D[t] = `/`(`*`(`+`(GS[t], TP[t], `-`(T[t]), `-`(PM[t]))), `*`(`+`(g[t], `-`(r[t])))) (5)
 


 

 

We now note that have 4 different cases:
 

 


1) Stable Equilibrium: Deficit and High Growth  →    and   


2) Stable Equilibrium: Surplus and High Growth  →    and   

 

3) Unstable Equilibrium: Deficit and Low Growth  →    and   

 

4) Unstable Equilibrium: Surplus and Low Growth  →    and   

 

 

 

 

High growth will always lead to a stable equilibrium where debt decreases over time. 

 

 

Low growth will always lead to an unstable equilibrium where debt increase over time. 

 


Deficit and high growth will lead to positive debt (deficit) where the fixed point is positive. 

 

 

Surplus and high growth will lead to negative debt (surplus) where the fixed point is negative.
 


 


 

We can illustrate theses 4 different cases as follows 

Case-1 Stable Equilibrium: Deficit and High Growth 

Case-2 Stable Equilibrium: Surplus and High Growth 

Case-3 Unstable Equilibrium: Deficit and Low Growth 

Case-4 Unstable Equilibrium: Surplus and Low Growth 

 

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