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Using a Sraffian price model we have that, $p=(1+r)(pA+wl)$ where w (nominal wage rate), r (average profit rate) are scalars; p (market prices), l (labor input coefficients) are vectors; A (producers' goods input coefficients) is a matrix. I wanted to prove that an increase in w would lead to a decrease in A or l (i.e., p is constant because of perfect competition). For that purpose I first try to refolmulate as $$A=\frac{p-wl(1+r)}{p(1+r)}$$ $$l=\frac{p(1-A(1+r))}{w(1+r)}$$

I take the derivates and I get,

$$\frac{\partial A}{\partial w} =- \frac{l}{p}<0$$ $$\frac{\partial l}{\partial w} =- \frac{p(1-A(1+r))}{w^{2}(1+r)}>0$$

Both results make no sense. An increase in wages should lead to an increase in machines (production goods) and a decrease in labor input. I could guess that an increase in wages would make companies invest in technological change so both A and l are reduced. I am sort of lost.

Thank you in advance

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  • $\begingroup$ You should clearly state which variable is endogenous and which are exogenous. If you have only one relationship and two endogenous variables $(A,l)$, nothing can be said regarding comparative statics. $\endgroup$ – Bertrand Aug 20 at 11:05
  • $\begingroup$ The implication that I find odd is that labor inputs are negative? If labor inputs are positive, then this rescales by some negative value and labor demand is decreasing in wages. $\endgroup$ – 123 Aug 21 at 0:10
  • $\begingroup$ If A is a matrix(as you claim) p and l must be vectors and the equations you derived for A and l make no sense. $\endgroup$ – Grada Gukovic Aug 21 at 23:22

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