Types od supply curve under competitive markets

Microeconomic theory presents eight types of markets, they are:

1. Competitive markets,
2. Monopoly market,
3. Oligopoly market,
4. Cartel market,
6. Monopolistic competition market,
7. Monopsony market,
8. Oligopsony market.

Each of these markets have their own characteristics, this post works on competitive market's supply curve. The main issue is to class supply curves according to short run and long run. We are going to start with

Maximun social surplus in competitive market

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.

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 = $\int_0^{0.819173}[\frac{1}{s+1}-s^{3}]ds - P.S = 0.148083$.

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. 

Making the time valuable on a raining day through calculation of FDI determinants for petroleum sector in Colombia

This note shows how standard profit maximization microeconomic problem can be taken in useful empirical model through econometric tools, therefore microeconomic theory taught in classrooms is proper. The main conclusion is petroleum foreign firms invest into colombian petroleum sector if profits is positive, oil crude production shows an increasing trend, labor cost is low, security environment shows improvements and oil crude price shows an increasing trend. This conclusion is so intuitive but making this effort to assess the validity and usefulness of microeconomic theory.

Firms in microeconomic theory problem can be set as

 $\underset{\{q\}}{Max} \ \ \pi=pq-wL-RK$
$s.a$
$q=F(K,L)$,

One can make this problem easier taking restriction into main problem and capital fixed $\overline{K}$ due to short run scenario, therefore:

 $\underset{\{q\}}{Max} \ \ \pi=pq-wL(\overline{K},q)-R\overline{K}\ \ \ (1.1)$

A. Firms will be in business if their profits reaches a minimum value

Firms will invest in this sector if profits $\pi$ is higher than $\overline{\pi}$, a special case is when $\overline{\pi}=0$, therefore the general case requires:

$pq-wL(\overline{K},q)-R\overline{K}\geq{\overline{\pi}} \ \ \ (1.2)$

therefore one can express a locus $(\overline{K},\overline{\pi})$ as

$(K,\pi})$ without bars, it means firms choose capital and profit they want to reach according to 1.2,

This is an important result to this note's propose due to data is available to fit it, Through first taylor expansion around long run optimal solution and under equality one can approximate 1.2 by:

$p(q-q^{E})-wL_{K}(K-K^{E})-wL_{q}(q-q^{E})-R(K-K^{E})=(\pi-\pi^{E})$
or
$K=\frac{\pi^{E}-q^{E}p-\pi-q(wL_{q}-p)+K^{E}R+K^{E}L_{K}w+q^{E}wL_{q}}{R+L_{K} w}$

an alternative compact form which is useful to fit data:

$\boldsymbol{K_{t}= \beta_{1} \pi_{t} +\beta_{2}q_{t}+ \epsilon_{t}}\ \ \ (1.3)$

this equation takes into account that through time there is a relation between $K_{t}$  and $\pi_{t}$. $p_{t}$ is not in 1.3 due to there is not empirical statistical evidence that capital as stock depends  directly  on this variable but one could use this fact into mathematical model. On the other hand, $ \pi_{t}$ shows a strong dependence of $p_{t}$ as it is showed below, therefore one takes into account $p_{t}$ indirectly .

It must be highlighted that above lines are not an optimization result.

B. Firms will produce q amounts under first order conditions optimal problem

Solution of 1.1 throughout optimization is a profits function as microeconomic textbooks point out:

$\pi=\pi(p,w,K) \ \ \ (1.4)$

as in the previous case the lineal form comes up throughout:

$\boldsymbol{\pi_{t}= \alpha_{1}p_{t} +\alpha_{2}w_{t}+ \alpha_{3}K_{t}+\mu_{t}} \ \ \ (1.5)$

C. Econometric model.

Equations 1.3 and 1.5 can be taken to fit the model, one can solve the long run model through envelope theorem applied to solution in 1.2 for $K$ but this is not the case, the target is find the relation between these two endogenous variables $(\pi,K)$ and exogenous variables $(q,p,w)$

Table 1 shows the output, conclusions are: first, profits oil foreign firms has a labor elasticity negative, real oil crude price elasticity positive, stock FDI into petroleum sector positive; second, the impact on FDI stock oil foreign firms has a profits oil foreign firms elasticity positive and oil local production elasticity positive; third, the  agency promotion of petroleum (ANH) and Plan Colombia show a strong significative relation with FDI as stock in petroleum sector, it means these two facts improved FDI into Colombia, they are not in the mathematical model due to they are idiosyncratic characteristics from Colombia.

Table 1. Oil crude FDI determinants 

(Annual data, three step OLS)

Variables	Log[Profits oil foreign firms]+	Log[FDI stock oil foreign firms]+
Log[labor remuneration]	-0.652* (0.115)
Log[Real oil crude price]	0.443* (0.443)
Dummy ANH	0.359 *** (0.224)
Log[Stock FDI petroleoum]	0.949* (0.048)
Log[Profits oil foreign firms]		0.488*  (0.094)
Log[Colombian oil production]		0.418*  (0.040)
Dummy Plan Colombia		0.583* (0.209)
R2	0.99	0.99
Probability Chi2	0.00	0.00
Observation number	41	41

              +Variables in natural logarithm; standard deviation (…).

              *p-value: less and equal than 0.01; ** p-value: less and equal than 0.05;*** p-value: less and equal than 0.1. **** p-value higher than 0.1.

                Source: own calculations. Stata 12.1.