This post is to show what is the volume of production to get the maximum social welfare under competitive market, the social welfare is measured through social surplus. The idea of this presentation comes from Mas-Colell, Whinston and Green 1995 chapter 10, pag 328-331. These authors give the main idea but they do not develop in detail the argument that is presented in this post.
Microeconomic theory takes consumer surplus and producer surplus as a measure of social welfare. There are two types of demand to evaluate the consumer surplus, they are Hicksian demand (compensated demand) and Marshallian demand, both demands show their own characteristics, moreover in some cases they are equal (see chapter 4 and exercise 4.19 from Jehle and Reny 2011). We are going to ignore this fact and we will take the consumer surplus as the area under demand curve (no matters what type is) until market price $\(p^{M}\)$ as figure 1 shows.
Microeconomic theory takes consumer surplus and producer surplus as a measure of social welfare. There are two types of demand to evaluate the consumer surplus, they are Hicksian demand (compensated demand) and Marshallian demand, both demands show their own characteristics, moreover in some cases they are equal (see chapter 4 and exercise 4.19 from Jehle and Reny 2011). We are going to ignore this fact and we will take the consumer surplus as the area under demand curve (no matters what type is) until market price $\(p^{M}\)$ as figure 1 shows.
Figure 1. Consumer surplus
For producers surplus, one can takes the area above of supply curve and below of market price $\(p^{M}\)$ as figure 2 shows. The area under supply curve can be taken as the total cost of produce "q" units.
2. Producer surplus
Therefore the total social benefit (consumer surplus plus producer surplus) can be calculated as the difference of total social benefit menus total social cost as figure 3 shows.
Figure 3. Social welfare (Total social surplus)
In terms of equations mean:
Demand curve: $p^{d}[q]$ where $\frac{dp^{d}[q]}{dq}<0$, $\forall{q}$
Supply curve: $p^{o}[q]$ where $\frac{dp^{o}[q]}{dq}>0$, $\forall{q}$
Total social benefit (SB): $\int_0^{q}[p^{d}[s]-p^{o}[s]]ds$.
Now, it takes attention where society can get the maximum social welfare, therefore one can use optimisation tools and get:
$SB^{*}= \underset{q}{Max}[\int_0^{q}[p^{d}[s]-p^{o}[s]]ds]$
The Firts Order Conditions (F.O.C) are:
$\frac{dSB}{dq}=0$, therefore,
$p^{d}[q^{E}]= p^{o}[q^{E}]$, it means the society gets the maximum social welfare where the competitive equilibrium is.
To be sure that $q^{E}$ is a maximum one takes the Second Order Conditions (S.O.C):
$\frac{d^{2}SB}{d^{2}q}$,
and one gets
$[\frac{dp^{d}[q]}{dq} - \frac{dp^{o}[q]}{dq}]<0$ as it is required for a maximum.
One can be interested in dividing total social benefit into consumer surplus (C.S.) and producer surplus (P.S.), therefore:
P.S = $P^{M}q^{E}-\int_0^{q^{E}}p^{o}[s]ds$
C.S = $S.B^{*}-P.S$.
Example:
Demand curve: $p[q]=\frac{1}{q+1}$, $\forall{q\geq{0}}$
Supply curve: $p[q]=q^{3}$, $\forall{q\geq{0}}$
The maximum social surplus is given by:
$SB^{*}: \underset{q}{Max}\{\int_0^{q}[\frac{1}{s+1}-s^{3}]ds\}$, therefore F.O.C lets,
$\frac{1}{q+1}=q^{3}$, the equilibrium is:
$q \approx 0.819173$, and $p\approx 0.5497$.
The S.O.C are:
$\frac{d^{2}SB}{d^{2}q}=\frac{-1}{(q+1)^2}-2q^{2}$, which is less than zero $\forall{q}$.
Now the optimum social surplus is:
$SB^{*}: \int_0^{0.819173}[\frac{1}{s+1}-s^{3}]ds=0.485807$.
The P.S. is:
P.S= $(P^{M}*q^{E})- \int_0^{0.819173}s^{3}ds=(0.5497)(0.819173)-\int_0^{0.819173}s^{3}=0.337724$,
and the C.S is:
C.S = $\int_0^{0.819173}[\frac{1}{s+1}-s^{3}]ds - P.S = 0.148083$.
2. Producer surplus
Therefore the total social benefit (consumer surplus plus producer surplus) can be calculated as the difference of total social benefit menus total social cost as figure 3 shows.
Figure 3. Social welfare (Total social surplus)
In terms of equations mean:
Demand curve: $p^{d}[q]$ where $\frac{dp^{d}[q]}{dq}<0$, $\forall{q}$
Supply curve: $p^{o}[q]$ where $\frac{dp^{o}[q]}{dq}>0$, $\forall{q}$
Total social benefit (SB): $\int_0^{q}[p^{d}[s]-p^{o}[s]]ds$.
Now, it takes attention where society can get the maximum social welfare, therefore one can use optimisation tools and get:
$SB^{*}= \underset{q}{Max}[\int_0^{q}[p^{d}[s]-p^{o}[s]]ds]$
The Firts Order Conditions (F.O.C) are:
$\frac{dSB}{dq}=0$, therefore,
$p^{d}[q^{E}]= p^{o}[q^{E}]$, it means the society gets the maximum social welfare where the competitive equilibrium is.
To be sure that $q^{E}$ is a maximum one takes the Second Order Conditions (S.O.C):
$\frac{d^{2}SB}{d^{2}q}$,
and one gets
$[\frac{dp^{d}[q]}{dq} - \frac{dp^{o}[q]}{dq}]<0$ as it is required for a maximum.
One can be interested in dividing total social benefit into consumer surplus (C.S.) and producer surplus (P.S.), therefore:
P.S = $P^{M}q^{E}-\int_0^{q^{E}}p^{o}[s]ds$
C.S = $S.B^{*}-P.S$.
Example:
Demand curve: $p[q]=\frac{1}{q+1}$, $\forall{q\geq{0}}$
Supply curve: $p[q]=q^{3}$, $\forall{q\geq{0}}$
The maximum social surplus is given by:
$SB^{*}: \underset{q}{Max}\{\int_0^{q}[\frac{1}{s+1}-s^{3}]ds\}$, therefore F.O.C lets,
$\frac{1}{q+1}=q^{3}$, the equilibrium is:
$q \approx 0.819173$, and $p\approx 0.5497$.
The S.O.C are:
$\frac{d^{2}SB}{d^{2}q}=\frac{-1}{(q+1)^2}-2q^{2}$, which is less than zero $\forall{q}$.
Now the optimum social surplus is:
$SB^{*}: \int_0^{0.819173}[\frac{1}{s+1}-s^{3}]ds=0.485807$.
The P.S. is:
P.S= $(P^{M}*q^{E})- \int_0^{0.819173}s^{3}ds=(0.5497)(0.819173)-\int_0^{0.819173}s^{3}=0.337724$,
and the C.S is:
C.S = $\int_0^{0.819173}[\frac{1}{s+1}-s^{3}]ds - P.S = 0.148083$.
Conclusion
This post showed where the maximum social surplus is got under competitive market. It can be interested for those who works on social welfare that come from competitive markets and for those who are interested in the methodology for getting the social welfare from any economic model. I will give the cases under taxes, monopoly and a FDI model in the next months. If you are interested and want collaborate follow me on twitter: @Humberto_Bernal.
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