A Math Error with Hazard Rate

 

A Math Error with Hazard Rate.hwp

 

JEL Classification: O30, C10

Keywords:   innovation, hazard rate

Abstract

Some economists treat hazard rate (λ) as an instantaneous probability. But λ is not a probability but a rate. Hazard rate is a decrease rate of the survival function. The survival function is the probability of no innovation until time t. As time goes, the probability of innovation success increases. The probability of the present technology's survival decreases. Hazard rate indicates how fast it decreases as time goes. The instantaneous probability at time t is not λ=f(t)/(1-F(t)) but f(t)=λ*S(t) that is the instantaneous decrease of S(t), the survival function.

 

1. Introduction

Dinopoulos and Segerstrom (AER 1999. p. 460, JIE 1999. p. 204), Segerstrom (1998. p. 1298), Aghion and Howitt (1998. p. 55.), Aghion and Howitt (1992. p. 329.), Barro and Sala-i-martin (1995. p. 249), Choi (1991. p. 599), Grossman and Helpman (1991. p. 48.), Grossman and Shapiro (1986. p.585.), Scotchmer and Green (1990. p. 137) regard hazard rate as the constant, perpetual, instantaneous probability.

But Kamien and Schwartz (1972. pp.46-47.), Loury (1979. p. 398.), Lee and Wilde (1980. p. 431.), Dasgupta and Stiglitz (1980. p. 15.), Mortensen (1982. p. 969.), Reinganum (1982. p.674.), Reinganum(1985. p. 85.), Dixit(1988. p. 319) and most other economists use S(t)*λ (hazard rate multiplied by the survival function) as the instantaneous probability.

2. Who are right?

Let's assume Poisson distribution the mean of which is λ. Then the probability of waiting until time t (no innovation until time t) is

   1 - F(t) = e^-λt.

The probability density function of innovation success time is

   f(t) = λe^-λt.

So hazard rate is

   f(t)/(1-F(t)) = λ.

Now can we use λ as the constant, perpetual, instantaneous probability? No, we cannot. The economists who use λ as the constant, perpetual, instantaneous probability are wrong.

1) Probability Density Function?

What is their probability density function? What is the time interval they consider? We need a time interval to get the probability from a probability density function. In the continuous model, without the time interval the probability is always zero. If their probability density function really were, then could we get 1 by integrating their probability density function?

They say the probability density function is constant λ. Then time is limited to 1/λ. Is it plausible?

2) What does f(t)/(1-F(t)) mean?

λ is hazard rate. Hazard rate is not a probability. What is hazard rate? Hazard rate is a decrease rate of the survival function. (Greene, 1997. p. 987., Lawless, 1982. p. 9., Cox and Oakes, 1984. p. 14.)

λ(t) =

If we assume Poisson distribution, then the probability density function of innovation success time is exponential distribution f(t) = λe^-λt. The survival function, the probability of waiting until time t (no innovation until time t), is 1-F(t)=e^-λt. And hazard rate is constant λ.

What does a decrease rate of the survival function mean? As time goes, the probability of innovation success increases. The probability of the present technology's survival decreases. Hazard rate indicates how fast it decreases as time goes. A growth rate of economy indicates how fast economy grows. Anyone does not insist that a growth rate is a probability. Hazard rate indicates how fast a certain probability decreases. But hazard rate in itself is not a probability but a rate.

3) The instantaneous probability is not f(t)/(1-F(t)) but f(t)

Hazard rate is f(t)/(1-F(t)). It is not a probability but a CONDITIONAL probability. Here 1-F(t) is condition. And the condition changes continually as time goes. Conditional probability density functions are totally different if their conditions are different. We cannot make a new distribution by selecting some portions from different distributions.

λ needs the condition that the innovation is not ready until t. If we want to use λ as "the constant, perpetual, instantaneous probability," we should always assume that the innovations is not ready until time t. This has a contradiction in itself. Let's think about two time instant, t=2 and t=3. They say the instantaneous probability at t=2 is λ. Also they say the instantaneous probability at time t=3 is λ. At time t=3, they need the condition 1-F(t). It means there is no innovation until time t=3. Then they should assume the instantaneous probability at time t=2 is 0. How can they solve this contradiction? The instantaneous probability at time t is f(t). Why do they ignore f(t)?

3. Conclusion

Grossman and Helpman (1991. p. 47.) say they mimic the models of Lee and Wilde (1980) and others. But they have mistake in mimicking the right models. The right models use f(t)=λ*S(t) not λ.

λ =

f(t)=λ*S(t)=λ*(1-F(t))

These equations will clear confusion. f(t) is the right probability. λ, hazard rate, is a decrease rate of the survival function. S(t), the survival function, is  the probability of waiting until time t (no innovation until time t). So f(t), the instantaneous probability, is S(t) multiplied by λ. f(t) is the instantaneous decrease of S(t).

References

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A Math Error with Hazard Rate.hwp

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