Which number counts officially as the " slope of the demand curve" : $\frac {\mathit d P(Q)}{\mathit dQ}$ or$\frac {\mathit d Q(P)}{\mathit dP}$?

Question

This is a very elementary question from a complete novice, and aiming first at preventing a possible misunderstanding. Thanks in advance.

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Suppose we have a linear demand curve with Y intercept $(0,20)$ and X intercept $(10,0)$. Which number counts as " the slope" of this curve in economics : $-1/2$ or $-2$?

In other words: by " slope" do we mean the gradient of the $P(q)$ function or the gradient of the $Q(p)$ function?

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Outside economics, the curve would be said to have a slope of :

$$\frac{\Delta y} {\Delta x} = \frac {10}{-20} = -\frac {1}{2}$$

and would be thought as a representation of the function

$$P(Q) = -\frac {1}{2} Q + 10$$

with Q ( = quantity) as independent variable and P (=price) as dependent variable.

However, it seems ( from some readings I've made) that the number denoted in economics by the expression " slope of the demand curve" is in fact the slope of the graph representing the following function :

$$Q(P) = -2P + 20$$

with Price as independent variable and Quantity as dependent variable

Answer 1

It is a gradient of $Q(P)$. $Q(P)$ is the demand function. $P(Q)$ is the inverse demand function. Even though confusingly when we plot demand we typically plot $P(Q)$, the demand function is actually $Q(P)$ since the function gives you output (demand) as a function of inputs (price). The inverse demand function inverts the function to get price from demand.

Note both $Q$ and $P$ are endogenously determined, so there are no exogenous variables here so one can plot either $P$ or $Q$ on y-axis.