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

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

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?

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

## 1 Answer

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.