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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?

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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.

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  • $\begingroup$ Thanks for the clear answer @AlecosPapadopoulos. It's "unfortunate" that this demand function implies that the optimal price $p^*\rightarrow0$ (with quantity going to infinity); it makes this demand function hard to justify for any practical application. $\endgroup$ – Ubiquitous Dec 4 '14 at 17:19
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    $\begingroup$ This happens because there is no cost. Even if for a range of quantity produced there may be some industries where one could argue that marginal cost is zero (e.g. Telcos, Software), once we start considering "very large" demand, you will necessarily have to consider increased costs (with maybe a nasty-looking step cost function). So price and quantity will be bounded away from the extremes, after all. $\endgroup$ – Alecos Papadopoulos Dec 4 '14 at 17:50

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