# Partial Derivative Problem ( Two Variable Function).

Two commodities $Q_1$ and $Q_2$ are said to be substitute commodities if an increase in the demand of either results in a decrease in the demand of the other.

Let $D_1(p_1,p_2)$ and $D_2(p_1,p_2)$ be the demand functions for $Q_1$ and $Q_2$ respectively, where , $p_1$ and $p_2$ are the respective unit prices for the commodities.

We need to check the signs of $\dfrac{\partial D_1}{\partial p_1}$ and $\dfrac{\partial D_2}{\partial p_2}$.

My thought on the problem :

$\dfrac{\partial D_1}{\partial p_1}$ tells us the change in demand($D_1$) with respect to its price($p_1$).

1). With the decrease in prices , the demand would eventually grow up , thus , $\dfrac{\partial D_1}{\partial p_1}>0$ ,

2). But , with the increase in prices , the demand($D_1$) would decrease and demand($D_2$) would increase , as given in the problem , thus $\dfrac{\partial D_1}{\partial p_1}<0$ and $\dfrac{\partial D_2}{\partial p_1}>0$

Similar argument can be given for $D_2$.

But the solution says , $\dfrac{\partial D_1}{\partial p_1}<0$.

How can we conclude that ? We're not supposed to consider the first case ? Could anyone tell , what am I doing wrong ?

• According to the law of demand, if the price of a commodity goes up then the quantity demanded will go down. This is because if the price of a good goes up then consumers will buy less of good $X$ and more of good $Y$, if the two goods are substitutes and if other factors are fixed. – OGC Aug 16 '15 at 6:17

If the two goods are substitutes then an increase in $p_{1}$ will decrease $D_{1}$, so $$\frac{\partial D_{1}}{\partial p_{1}}<0.$$ On the other hand, an increase in $p_{1}$ will increase $D_{2}$, so $$\frac{\partial D_{2}}{\partial p_{1}}>0.$$
Similarly, an increase in $p_{2}$ will decrease $D_{2}$, so $$\frac{\partial D_{2}}{\partial p_{2}}<0.$$ On the other hand, an increase in $p_{2}$ will increase $D_{1}$, s0 $$\frac{\partial D_{1}}{\partial p_{2}}>0.$$