# Interpretation of market condition given by relation between elasticities

The total market demand is given by the sum of the two destinations ($$e$$ and $$f$$) given to the product.

Given these conditions:

• Demand: $$d(p) = e(p) + f(p)$$ where:

• $$d(p) > 0$$, $$d'(p) \leq 0$$;
• $$e(p) > 0$$, $$e'(p) \leq 0$$;
• $$f(p) > 0$$, $$f'(p) \leq 0$$;
• Supply: $$s(p)$$ where $$s(p) > 0$$ and $$s'(p) \geq 0$$;

• Equilibrium: $$d(p) = s(p)$$;

How can the condition below be interpreted?

• $$\frac{f'(p)}{f(p)}-\frac{d'(p)}{d(p)}> \frac{e(p)}{f(p)}\frac{s'(p)}{s(p)}$$

It may be helpfull to see that's possible to multiply by $$p$$ and get some elasticities:

$$p*\big(\frac{f'(p)}{f(p)}-\frac{d'(p)}{d(p)}> \frac{e(p)}{f(p)}\frac{s'(p)}{s(p)}\big) \Rightarrow$$ $$\varepsilon_{f}-\varepsilon_{d} > \frac{e(p)}{f(p)}\varepsilon_{s}$$

This is a simplification of the whole situation from the The Ethanol-subsidy Multiplier example (or notebook file).

• Well i dont see how you postulate d(p)=s(p) with d(p)<0 and s(p)>0. – Armenthus Jan 10 at 22:29
• Indeed, @Armenthus. There was was a typo, although that relation was still possible (at least for a p*). – Rodrigo Remedio Jan 11 at 15:33