The true about crude oil price decline impact on economic welfare

Colombian government and private firms broadcast biased information about crude oil price decline. They broadcast the government income reduction because of lower royalties, lower utilities from Ecopetrol (petroleum government-private firm) and lower make value of petroleum firms. Public opinion needs to know the benefits from this reduction. These benefits are lower final good prices and lower exchange rate in the long run (after 9-12 months from negative shock on crude oil price). Therefore, government has to work on meeting these benefits as soon as possible to take advantage of this crude oil price decline. The fist step is to regulate the gasoline price for load transportation such as public transport and trucks; moreover, they have to increase taxes for familiar cars to regulate cars' pollution. 

Author: Humberto Bernal,  

Economist,

e-mail: zhumber@gmail.com

Twitter: Humberto_Bernal

It can be download @ PDF

This note deals with crude oil price decline and its impact on nominal prices in Colombia. There is a biased talking about the impact on economic growth and economic development because of crude oil price decline in the last nine months. This talking points that crude oil price decline comes with costs in terms of lower government income, higher exchange rate (Col$xxx per US$1.00) and petroleum companies lower market value. However, it is a part of this debate, so  crude oil price decline will bring a reduction on final goods prices also, and it means better social welfare.

To point this long run benefit, I take a time series model called VEC that takes into account Crude Oil Price (WTI), Colombia Consumers Price Index (Col_CPI); Colombian Whole Price Index (Col_WPI), Colombian Nominal Exchange Rate (Col_ER) and the Unites States Consumer Price Index (USA_CPI). These five variables come in monthly span since January of 1986 to December of 2014; moreover, they are lagging 12 months each  one to get seasonally effects. This model is robust in statistical terms.

Results from this model goes with social welfare after the shock. Through impulse response analysis, results show that first 9-12 months after negative shock (-1.0% decline on crude oil price), the Colombian CPI increases; however, after these 9-12 months, Colombia CPI shows a decline; therefore, it means final consumers will enjoy a higher power of purchasing. The Colombia Nominal Exchange Rate will increase (Colombian peso will face a devaluation against dollar) the first 9-12 months, but it will decline after this period, then it will converge to the long run level without any change. The Colombia Whole Price Index will show a decline. Finally, the Unites States Consumer Price Index will face a decline also as figure shows.  

In conclusion. Government economic authorities and private research firms in Colombia are broadcasting partial information about the effect of crude oil price decline. Of course, there are costs as they point through reduction of government income and petroleum firms market value. However, these people do not talk about benefits, and this document points them, so Colombia CPI, WPI and Nominal Exchange Rate will show a decline after 9 - 12 moths from initial negative shock on crude oil price. It means that Nominal Exchange Rate short run movements are transitory (speculation), so after 9 months the Colombian Nominal Exchange Rate will come back to its long run equilibrium; nonetheless, it can spent more time to get its long run value again due to negative shock is not transitory, it is permanent in the last 9 months, but prices will decline by sure in the next months if government works on gasoline price.

General Equilibrium Model 2x2 under Cobb-Douglas functions

Author: Humberto Bernal,  

Economist,

e-mail: zhumber@gmail.com

Twitter: Humberto_Bernal.

It can be download @PDF

This note is to broadcast my research on General Equilibrium Model under two firms and two consumers GE 2x2. There are huge of information about it such as Debreu (1972), Arrow and Hahn (1984), Arrow and Intriligator (1986) and text books such as Kreps (1990), Varian (1992), Mas-Colell, Whinston and Green (1995) and Jehle and Reny (2011), Nicholson and Snyder (2011).  However, they do not show how to solve this type of models in detail, they work through generalizations such as Arrow, Hahn and Debreu did. Therefore, the contribution of this document is to work on this GE 2x2 through uses of values; moreover, at the end of this contribution, I found a novel theorem about relative prices under trade.

To show how GE 2x2 works, there are three models taken into account. First, it is a model under diseconomies of scale; second, it is a model under constant returns to scale; finally, it is the Heckscher-Ohlin model. Solution of these models can be interesting for those who want to understand in easy way GE 2x2 models such as researchers, teachers and students. Moreover, a novel theorem about relative prices under trade is presented at the end of document. This theorem is important because says that in the long run (the Heckscher-Ohlin model) there will not be labour and capital migration between countries no matter political restrictions or transport restrictions because of relative prices are equal between countries; it happens because at end all countries will enjoy equal technology for producing goods; moreover, internet can be taken as media to broadcast this technology.

In this research you will find how to plot Possibility Production Frontier and Contract Curves as figure 1, 2, 3 show.

Figure 1. Production Possibility Frontier GE 2x2

Figure 2. Production Possibility Frontier HO model Autarky

Figure 3. Production Possibility Frontier HO model Trade

Hopefully, the myth about GE 2x2 and its weaknesses can be mitigate although I am aware of its restrictions such as scenarios of market failures.

Bernal, Humberto. 2014. "General Equilibrium Model 2x2 under Cobb-Douglas functions". Public Access. 

The optimum tax under competitive market and diseconomies of scale

This note deals with optimum taxes under competitive markets and diseconomies of scale. The demand curve come from consumer theory and they can be Marshallian or Hicksian demand, they are in some cases equal (under quasilinear preferences).  We going to ignore the type of demand, the main issue concerned is the consumer welfare can be measure through consumer surplus. The market demand  of good j comes from summation of individual demands of each i which are equal for every consumer but not in income distribution, it means:

$Q^{d}(\bf{p},y)=\sum^{N}_{i=1}q_{i}(\bf{p},\alpha_{i}y)$

This market demand $Q^{d}(\bf{p}, y)$ does not change if income distribution changes see Jahle and Reny (2011) exercise 4.1 pag 188.  Now the point is to work with inverse demand, as this market demand function is onto and one to one for $p_{j}$, then its inverse exits. Ma-Colell, Whinston and Green (1995) deals this issue in detail pag 116 and Jahle and Reny (2011) pag 505.

Firms are in their long run production that means no fixed costs in their cost structure, then we can calculate the total cost (C) through marginal cost (MC) integration:

$C=\int \limits_0^{q_{j}}MC[s]ds$

As theory of firm points, the supply curve of firm h comes from $CM_{h}(q)=p_{j}$, then the market supply can be written:

$Q^{o}_{j}[\bf{p}]= \sum_{h=1}^{H}q_{h, j}(\bf{p})$

The above lines is an attempt to aggregate consumers consumption of good j and firms production of good j. Now with this market demand and market supply the tax t ($ per unit) has to be charged. Consumer and firms are interested in getting the maximum welfare, then they will maximize:

$SB^{*}=\underset{Q}{Max}[\int_0^{Q}[p^{d}[s]-p^{o}[s]]ds-tQ]$

The Firts Order Conditions (F.O.C) are: 

$\frac{dSB}{dQ}=0$, therefore,

$p^{d}[Q^{E}]- p^{o}[Q^{E}]=t$,       E.1

It means the society gets the maximum social welfare where the difference between consumers price and firms price is equal to tax "t".

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

$[\frac{dp^{d}[Q]}{dQ} - \frac{dp^{o}[Q]}{ds}]<0$   $\forall {Q}$   as it is required for a maximum.

There is an interesting question about what is the "t" that maximizes the consumer and firms welfare?, in other words what is the tax that maximize the change in society welfare. To answer this question we follow few steps:

The solution above E.1 can be written as $Q^{E}=Q[t]$,

The initial state is with tax equal to zero, then we start in $Q^{E}=Q[0]$

The final state is $Q^{E}=Q[t^{*}]$ where we need to find the $t^{*}$ that makes the maximum positive change between these two states, then the optimisation problem is:

$SBT^{*}=\underset{t}{Max}[\int_{Q[0]}^{Q[t]}[p^{d}[s]-p^{o}[s]-t]ds$

It is fair to work on the last expression a bit. It is the maximum difference between social welfare under optimal tax $t^*$ and social welfare under tax equal to zero, in this case a negative difference, positive difference or null difference make sense. As the problem is solved under optimisation tools see Chiang, A (1992) pag 29-32, one gets:

$\frac{dSBT}{dt}=0$, therefore

$\frac{dQ}{dt}[p^{d}[Q[t^{*}]]-p^{o}[Q[t^{*}]]-t^{*}]-\int_{Q[0]}^{Q[t^{*}]}ds=0$

$\frac{dQ}{dt}\underline{[p^{d}[Q[t^{*}]]-p^{o}[Q[t^{*}]]-t^{*}]}-({Q[t^{*}]}-{Q[0]})=0$

The underline text is always zero no matter the tax value, see above problem E.1 and Post of "Maximum social surplus in competitive market - November 17 of 2012 wrote by me". Therefore,

${Q[t^{*}]}={Q[0]}$,

We got the best tax for society is $t^{*}=0$ due to loss of welfare under positive tax. Figure 1 shows the result.

Figure 1. Maximun social welfare under any tax and optimum tax

With this result the government is not agree due to she does not get any revenue, then she optimises her revenue $R[t]=tQ[t]$ no matter the social welfare value. Moreover, government knows that $\frac{dQ[t]}{dt}<0$, then

$\frac{dR[t]}{dt}=0$

$Q[t]+\frac{dQ[t]}{dt}t=0$

$[\frac{dQ[t]}{dt}][\frac{t}{Q[t]}]=-1$

Therefore the optimum tax for government is where tax elasticity of the volume is equal to -1. The existence of this maximum depends on volume function form.

Special case "The Laffer Curve"

The Laffer Curve is an interesting economic thought about the optimum tax see Laffer Curve. The next lines I try to set up this curve under above results.

Demand:

$P^{d}[Q]=a-bQ$   where $a>0$ and $b>0$.

Supply:

$P^{o}=\alpha Q$  where $\alpha >0$.

$P^{d}-P^{o}=t$

$a-bQ- \alpha Q =t$

The solution is

$Q[t]= \frac{a-t}{b+ \alpha}$   as one can see a must be greater than t, it is normal due to demand curve intercepts on P axis.

This optimum volume $Q[t]$ follows above results. Now the government revenue is:

$R[t]=t[\frac{a-t}{b+ \alpha}]$

But government optimizes for "t", then

$ \frac{dR[t]}{dt}=0$,

the solution is

$t^{*}= \frac{a}{2}$

This is where the tax elasticity of volume is equal to -1.

Now it is a maximum due to

$ \frac{d^{2}R[t]}{d^{2}t}<0$   $ \forall t$,   Figure 2 shows the result

Figure 2. Laffer curve a special case of optimum tax

The conclusion is a tax greater than 0 lets a lower optimum volume than tax equal to zero under diseconomies of scale, it means a positive tax lets getting Allocative Inefficiency. However, government has to get revenues from market transactions, therefore she looks for optimum tax that lets getting maximum revenue and she found an optimum tax where tax elasticity of volume equal to -1 but in some cases there is not solution, it depends on volume function characteristics. I showed a special case where the result is Laffer curve.

Those who want comment this post, here is my twitter: @Humberto_Bernal.

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.