Amazing Logarithm

Log-linearizing is a most common way to deal with data in economic study, perhaps in any data analysis. In my first macroeconomic and econometric class this semester, I encountered this widely used way of data transformation.
Common Reasons for taking log of the data:
1. Primary reason is to smooth the data, so as to show the patterns.
In real life, the data we collected, especially large sample of data from a big pool, have huge difference of their magnitudes.  Once we take log of the data, without changing the properties of the data, it helps us to spread the points more uniformly in the graph.

Let me illustrate with a simple sample of world per capita GDP, which exists big gap between different years. As we can see from the following two graphs, the data spreads more smoothly in the second graph. Without taking the log of the GDP, graph 1 clearly ignores the changes between 1000 and 1800.

 Figure 1: World per Capita GDP

Figure 2: Log of World per Capita GDP

2.Good properties

	Divided by 10	Original Data	Multiplied by 10
Data	0.1*A	A	10A
Log of Data	LogA-1	LogA	LogA+1

	Divided by 10	Original Data	Multiplied by 10
Change of Data	0.1	1	10
Change of Log Data	-1	0	1

Above tables show again the effect of smoothing data when you take log of the data. 

"The most important feature of the logarithm, which makes it so useful in analyzing data, is that, relatively, it moves big values closer together while it moves small values farther apart. This is seen in the picture to the left, where the original values are on the upper scale and their common logarithms are on the lower scale. In the log scale, small values that were close in the original scale (say, 1 and 10) are now the same distance apart as large values that were farther apart in the original scale (say, 10 and 100). " Source: http://www.jerrydallal.com/LHSP/logs.htm

(Thanks to Gerard E. Dallal)

Concluded from Gerard E. Dallal's Study, a logarithmic transformation will help us to make the original data with more appealing properties, such as symmetric, homoscedasticity and linearity. In one word, logarithm makes life simpler and easier.

3. Doesn't necessarily need to be linear. But performs very well for growth rate ( any functions with exponential or power ). For example, Cobb-douglas production function. Yt=(1+g)t*(1+)t*yt. After log transformation, you can easily see the linear relationship between log Y and time.


Some visual impression of logarithms:
-Logarithm in real life:
Logarithmic spiral

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Hope I can understand logarithm and data transformation deeper in the end of this semester.
Stay tuned.

Telecommunication Network Access Pricing Under Vertical Integration

When you make a long-distance phone call, it has to go through the local exchange carrier's local loop, switches, transmission facilities at both originating and terminating ends and through a trunk line in between the two points. Since the breakup of AT&T in 1984, competition in long-distance market has become a normal pattern in telecommunication industry. However, incumbent in the local markets with the essential facilities like local loop, or more precisely the very last mile form the local exchange to the end-user's home, has a major market power in the market.  Entrants in the long distance market have to pay access fee to get access to the local network to originate and terminate a call. 

Under vertical separation, bottleneck suppliers don't compete with the entrants in the long-distance market. The competition degree of the downstream market plays a important role in determine the market efficiency. Under vertical integration, incumbents share the long distance market with new entrants by providing them access to local network. They have to consider the tradeoff between retail sales to end-users and access fee gain from entrants when they set the access fee. Normally, there are four pricing rules of access fee, ECPR, Ramsey Pricing, Cost-based pricing and Global price caps.

 I. ECPR(Efficient Component Pricing Rule)

Access price is set equal to marginal cost of access plus the opportunity cost of access. The opportunity cost of access is the forgone profit contribution by selling access to downstream rather than selling to retail end-users. Here we assume the customer only subscribe one carrier.

               Pa=MCa+Copp= P1-C1. 

Here P1 is long-distance price of incumbent, C1 is the long-distance marginal cost of the incumbent. 

ECPR is relatively simple and it ensures that only entrance who can undercut the incumbent will enter. Also, "entry is neutral regarding operating profit for the incumbent", so it won't have incentive to destroy the entrants by predatory pricing, which is a potential problem of global price cap. However, just because the ECPR rule is simple, it is not suitable when products or services are differentiated. 

II. Ramsey Pricing

When maximize social welfare under the participation constrain, we can always get Ramsey pricing outcome. The mark up of local call prices, incumbent long-distance calls and entrants' long-distance calls are all obey the inverse elasticity rule. Here we use superelasticities to include the substitution and complementarity among different calls. The optimal access price under Ramsey pricing is composed of two parts, which are the standard Ramsey pricing equation and the substitution effect of the incumbents' sales of local network and their loss of retail sales. 

a = 2C0 + λ/(1+ λ)(p2/ε2) + δ(p1 – 2C0 – C1)   (Armstrong, Doyle and Vickers, 1996) 

Also, 2C0 +  δ(p1 – 2C0 – C1)  is called sophisticated ECPR.  δ here is a displacement ratio, which expressed the reactions of competitors. It is the ratio of quantity reaction of incumbent to the price change of entrant's long-distance call price and quantity reaction of entrances. Sometime, we call it business-stealing effect. When δ =1, Pa= P1-C1, the simple ECPR. 

The universal drawback of Ramsey pricing is that the requirement of information is high, and the computational burden is heavy.

III. Cost Based Pricing- Long-run Incremental Cost Pricing( LRIC Pricing)

"http://www.ictregulationtoolkit.org/en/Section.2092.html"

IV. Global Pricing Cap

Global pricing cap is a price cap includes the access pricing into the multi-product price cap. It treats access product as one of the whole products into regulation. Under this method, if regulator can assign an idealized or perfect weight to the price cap which can lead to all product pricing at Ramsey pricing. Then the global pricing cap give a more flexible way for firms to reach the optimal pricing. However, just as we said, to calculate the optimal quantity weight is an ideal idea, which requires full information. Therefore, as a good theoretical method, global pricing cap never implemented. Moreover, applying a price cap may in a way enhance the incentive of incumbent to introduce the predatory pricing strategy. The local exchange carriers might set a price below the marginal cost to get rid of competition in the first stage, and recoup the revenue from monopoly pricing later. 

All of the above four access pricing method is designed for ONE WAY ACCESS situation, we will discuss later about two-way access and other topics in network industry regulation.

Ratchet Effect

Ratchet effect is an effect when we discuss about regulation lag or the issue raised when we talked about commitment problems.  Any finite contract with asymmetric information actually might exhibit the ratchet effect.

For example, under Bayesian mechanism, regulator at first only know the information of regulated firms based on the distribution information. At stage one, if there is two options, a high power incentive and a low power incentive, regulated firms will automatically choose the one favored to themselves and as a result reveal the type of the firm. Self-selection is an important feature of bayesian mechanism. At stage 2, usually when the first contract expires, regulator can use the information of firms' ex post cost to reset the firm's prices. Without commitment, firms are in a situation of " punished by doing well in the last period." If firms know about this mechanism they will have a less incentive to engage in cost reducing effort.  It can leads to a series of problems, less investment, cost inflation, etc.

What if there is a commitment of the regulator of not changing the contract for a relatively long period (which still has the ratchet effect in the last period). It can reduce the above problems, but as all long-term contracts it have the problem of inflexibility . In a world changed almost every second, a fixed contract can not probably deal with the turmoil, say policy and technology changes. Not to mention the unredeemed promise. In a regulatory world, we can't just say, hey, don't cry for the pouring milk. We have to correct the mistakes.

Joskow mentioned in "Regulation of natural monopolies"that "the behavior of a firm will depend on the information that its behavior reveals to the regulator ex post and how the regulator uses that information in subsequent regulatory reviews." It is a game between the regulator and regulated firms. When there is no commitment, we can analysis the situation happened from both sides' behavior. He cited the UK's RPI-X mechanism as a real world example. The RPI-X mechanism is a price cap adopted an annual inflation rate and a target productivity change factor"X", which will be reset every 3-5 years to reflect the current realized cost of service. Under the observation, firms under this regulation made their greatest cost reduction efforts in the earlier period and reduce the effort as the review approached.

Dilemma raised, solutions must be offered. Usually, there are two ways out. First, rely on the law. Because even without commitment , firm have the protection of law. So it can partly increase the confidence and stability of investments. Second approach, which is kind of meaningless, is abandon the incentive regulation, just use the rate of return regulation.

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reference:
  Paul L. Joskow, Regulation of Natural Monopolies
For more information:
1. Laffont and Tirole,(1993), A theory of Incentive in Regulation and procurement
2. Laffont and Tirole, Competition in Telecommunications