Demand Function for $X$ is $$ Q_x=420 - 7P_ye^{P_x} + 0.06Y^2+390P_y^3 $$
Find price, cross and income elasticites of demand at $P_x = 7.1$, $Y=1764$, $P_y = 8.2$
Explain.
Here is my workings:
$$ Q_x = 332,588$$
$$ \epsilon_{px} = -7 * \frac{7.1}{332588} = -0.000149 $$
$$ \epsilon_{py} = 390 * \frac{8.2}{332588} = 0.00938$$
$$ \epsilon_y = 0.06 * \frac{1764}{332588} = 0.00318 $$
When taking $Y$ value, would i have to square root it? As it is $ 0.06Y^2$, not $ 0.06Y $ Likewise for $P_y$?
Thanks.
Hint: Your differentiation went wrong.
$$\huge{\epsilon}_{\normalsize{P_x,Q_x}}\normalsize{=\frac{\partial Q_x}{\partial P_x}}\cdot \frac{P_x}{Q_x}$$
$$\frac{\partial Q_x}{\partial P_x}\cdot P_x=-7\cdot P_y\cdot e^{P_x}\cdot P_x$$
If you differentiate w.r.t. $P_x$ then $P_y$ is treated as a constant. And the derivative of $e^{P_x}$ w.r.t $P_x$ is just $e^{P_x}$. Thus
$$\huge{\epsilon}_{\normalsize{P_x,Q_x}}\normalsize{=-7\cdot P_y\cdot e^{P_x}\cdot P_x}\cdot \frac{1}{420 - 7P_ye^{P_x} + 0.06Y^2+390P_y^3}$$
Inserting the values
$$\huge{\epsilon}_{\normalsize{P_x,Q_x}}\normalsize{=-7\cdot 8.2\cdot e^{7.1}\cdot 7.1}\cdot \frac{1}{420 - 7\cdot 8.2\cdot e^{7.1} + 0.06\cdot (1764)^2+390\cdot (8.2)^3}=-1.485$$