<p>We have that </p>

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

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

<p>So</p>

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

<p>or</p>

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

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

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

<p>So we have that the demand function the OP seeks must satisfy</p>

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

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

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

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

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

CC BY-SA 3.0
2014-12-04T17:19:16.150Z
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.
Ubiquitous
CC BY-SA 3.0
2014-12-04T17:50:43.020Z
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.
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-04T15:32:25.833Z
2014-12-04T15:32:25.833Z
Alecos Papadopoulos
<p>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 $$D_1(p^*;\mathbf{a})p^*+D(p^*;\mathbf{a})=0.$$</p>

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

<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?</p>

<microeconomics><monopoly><consumer-surplus><supply-and-demand><producer-surplus>
Is there a class of demand functions that deliver equal surplus to consumers and a monopolist?
Ubiquitous
# Is there a class of demand functions that deliver equal surplus to consumers and a monopolist?

(Asked by: username_0 on 2014-12-04)

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?

**Tags:** microeconomics, monopoly, consumer-surplus, supply-and-demand, producer-surplus

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## 1 answers

### Answer by username_1 on 2014-12-04. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_0**: 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.  
> **username_1**: 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.  

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