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Solving Kaldor's (1957) Model

Equations

(1) Capital Stock "desired" by firms:

(2) Investment Function:

(3) Saving Function:

(4) Technical Progress Function: , where or
(4')

Dividing (2) by :
(2')

From (1) we may rewrite: , from (4') we have , and from (4) . By using these equations we can rewrite (2') so that

Suppose that the macroeconomic equilibrium holds at the end of each period () and that the workers "spend what they earn" , then we have:
(5)

And then dividing (5) by we have or

Thus,

Finally solving   (and renaming a=and b=):

(6)  =
As we may notice (6) is a nonlinear difference equation.

Numeric Solution

Parameters

Proportion out of output held as capital (see equation 2)

Productivity's autonomous growth () + 1

Proportion of profit share

Sensibility of technical progress to the capital accumulation process

Saving out of profits ratio

Initial investment ratio

This creates an array of values to be use in iterations

It fills the array with the results of the recursive equation (6) which defines the investment ratio over time.

Number of Periods: 15

Steady-State values (Numeric Solving)

Productivity Growth Ratio

Capital-Output ratio (v)

Profit Share (Cambridge Equation)

Share of profits in income

Note that the necessary conditions with regard to the slope of the investment and savings curve are held:
is the output-capital ratio, which is a tantamount to the inverve of the capital-output (v)

Steady-State values (Algebraic Solving)

THE VALUES BELOW ARE THE SAME OF THE NUMERIC SOLVING ABOVE. IT INDICATES THAT BOTH RESULTS ARE CONSISTENT.

Productivity Growth Ratio

Capital-Output ratio (v)

Profit Share (Cambridge Equation)

Share of profits in income

References

Kaldor, N. (1957). “A Model of Economic Growth”. Economic Journal, Vol. 67.

By Fabio Hideki Ono - http://fhono.conjuntura.com.br