Is there a class of demand functions that deliver equal surplus to consumers and a monopolist?

Question

Consider a market with a monopolist firm that has zero marginal cost and faces demand $D(p;\mathbf{a})$, where $\mathbf{a}$ is a vector of parameters and $p$ is the price. The monopolist maximizes profit by solving $$\max_p D(p;\mathbf{a})p,$$ so that the optimal price, $p^*$, satisfies $$D_1(p^*;\mathbf{a})p^*+D(p^*;\mathbf{a})=0.$$

This optimal price, $p^*$, results in consumer surplus $$\text{CS}=\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp,$$ and producer surplus $$\text{PS}=D(p^*,\mathbf{a})p^*.$$

My question is: is there a family of demand functions, $D(p;\mathbf{a})$, such that $\text{CS}=\text{PS}$ always holds at $p^*$, and if so what does the functional form look like?

Answer 1

We have that

$$D(p^*,\mathbf{a}) = -\frac {d}{dp^*}\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp,$$

$$\Rightarrow \text{PS}(p^*) = -\text{CS}'(p^*)p^* \tag{1}$$

So

$$\text{PS}(p^*)= \text{CS}(p^*) \Rightarrow -\text{CS}'(p^*)p^* = \text{CS}(p^*)$$

or

$$\text{CS}'(p^*) + \frac 1{p^*}\text{CS}(p^*)=0 \tag{2}$$

which is a first-order linear homogeneous differential equation in $p^*$ with variable coefficent. Its solution is

$$\text{CS}(p^*) = B\exp\left\{-\int \frac 1{p^*}dp^*\right\} = B\exp\left\{-\ln p^*\right\}=B\frac 1{p^*},\;\; B>0 \tag{3}$$

So we have that the demand function the OP seeks must satisfy

$$\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp = B\frac 1{p^*} \tag{4}$$

Since it should hold $\forall p^*$ we can consider the derivative w.r.t to $p^*$ on both sides, to obtain

$$D(p^*;\mathbf{a}) = B\frac 1{[p^*]^2} \tag{5}$$

But since, again, it should hold $\forall p^*$, it holds $\forall p$. So

$$\text{PS} = \text{CS} \Rightarrow D(p;\mathbf{a}) = B\frac 1{p^2} \tag{6}$$ Verification of $(6)$ is straightforward.