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I was wondering, in a log-log model of output, labour and commodity price where output is the depedent variable, does it make sense if the coefficient on the price variable is interpreted as the "price elasticity of output"? Assume output is measured in constant monetary unit.

Let's say $log(gdp_t) = \beta_0 + \beta_1log(labour_t) + \beta_2\log(commodityprice) + \epsilon_t$

I've not seen $\beta_2$ being defined as the elasticity of output with respect to price. But, could this be correct at all?

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  • $\begingroup$ How does this log-log model looks like ? $\endgroup$ Jun 4, 2016 at 8:28
  • $\begingroup$ Edited the question above. $\endgroup$
    – london
    Jun 4, 2016 at 14:09

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I use some abbreviations and drop the index t.

$\ln(G)=\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon$

Taking both sides as an exponent of e

$e^{\ln(G)}=e^{\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\beta_2\ln(P)}$

$e^{\beta_0+\beta_1\ln(L)+\epsilon}=C$ is a constant if you differentiate w.r.t $P$.

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\ln(P^{\beta_2})}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}$

$\frac{\partial G}{\partial P}=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}$

The elasticity is $\large{\epsilon}\normalsize{=\frac{\partial G}{\partial P}\cdot \frac{P}{G}}$

$\large{\epsilon}=\normalsize{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}\cdot \frac{P}{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}}}$

=$C\cdot \beta_2\cdot P^{\beta_2 -1}\cdot\frac{P}{C\cdot P^{\beta_2}}$

You can finish by simplifying the term.

In general it is true that

$\large{\epsilon}=\normalsize{\frac{\partial \ln(G)}{\partial\ln(P)}}$

Thus the short way is to differentiate $\ln(G)$ w.r.t $\ln(P)$.

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  • $\begingroup$ thanks. But I wonder if it actually makes sense in the context of economics. Mechanics is fine, it is defined as elasticity, but in economics, I've not seen a text mentioning something along the lines " elasticity of output with respect to price". $\endgroup$
    – london
    Jun 5, 2016 at 14:11
  • $\begingroup$ Do you have a source for this model ? $\endgroup$ Jun 5, 2016 at 14:17
  • $\begingroup$ that's the problem, this is just a simple linear model to check the effects of commodity price changes on GDP. For instance, with labour, it is correct to say 'elasticity of output with respect to labour', this is the share of labour in output. But with prices, does it work in the same way? Price elasticity of demand and supply is another textbook case, but was wondering if it makes sense to say $\beta_2$ 'price elasticity of GDP'. $\endgroup$
    – london
    Jun 5, 2016 at 14:44
  • $\begingroup$ If we further simplify the equation it is $Y=L^{\beta_1}\cdot P^{\beta_2}$. Usually it is $C$ (capital) instead of $P$. But beside this we have to check if it makes sense that if the price level raises then the gdp raises. For a moment I would say no. I assume $\beta_2$ is positive. $\endgroup$ Jun 5, 2016 at 15:00

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