<p>I use some abbreviations and drop the index t.</p>

<p>$\ln(G)=\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon$</p>

<p>Taking both sides as an exponent of e</p>

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

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

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

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

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

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

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

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

<p>=$C\cdot \beta_2\cdot P^{\beta_2 -1}\cdot\frac{P}{C\cdot P^{\beta_2}}$</p>

<p>You can finish by simplifying the term.</p>

<p>In general it is true that </p>

<p>$\large{\epsilon}=\normalsize{\frac{\partial \ln(G)}{\partial\ln(P)}}$</p>

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

CC BY-SA 3.0
2016-06-05T14:11:03.047Z
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".
london
CC BY-SA 3.0
2016-06-05T14:17:49.217Z
Do you have a source for this model ?
callculus42
CC BY-SA 3.0
2016-06-05T14:44:54.237Z
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'.
london
CC BY-SA 3.0
2016-06-05T15:00:52.470Z
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.
callculus42
CC BY-SA 3.0
2016-06-04T14:58:27.913Z
2016-06-04T14:58:27.913Z
callculus42
<p>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.</p>

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

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

CC BY-SA 3.0
2016-06-04T08:28:14.497Z
How does this log-log model looks like ?
callculus42
CC BY-SA 3.0
2016-06-04T14:09:10.397Z
Edited the question above.
london
<macroeconomics><microeconomics><econometrics><regression>
price elasticity of output, makes sense?
london
# price elasticity of output, makes sense?

(Asked by: username_0 on 2016-06-03)

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?

**Tags:** macroeconomics, microeconomics, econometrics, regression

> **Comments:**
> **username_1**: How does this log-log model looks like ?  
> **username_0**: Edited the question above.  

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## 1 answers

### Answer by username_1 on 2016-06-04. Score: 1

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 $\ln(P)$.

> **Comments:**
> **username_0**: 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".  
> **username_1**: Do you have a source for this model ?  
> **username_0**: 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'.  
> **username_1**: 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.  

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