When you say, “Someone might use r(t) = . . . to value resources . . .” and then refer to the equation “r(t) = . . .” as a “valuation scheme,” you lead me to believe that you believe that r represents utility. But that is not how John is using r. In John’s section on net present value, John has postulated that the utility experienced by our plucky entrepreneur at time t == u(t) == a / r ^ t where a and r do not vary over time.
Although John did not explicitly mention the function u, I do so now because you seem to have confused u with r.
You wrote that r can very over time “without self-contradiction”, to which I replied, “not without our plucky entrepreneur becoming a money pump,” which I still believe to be the case. Of course, John’s model does not capture the full complexity of the choices and constraints facing an entrepreneurial software developer, but there is a good reason why most treatments of net present value assume that the discount rate does not vary over time.
I did not misunderstand. The discount over a time period dt with a constant r is 1/r^dt. If we want a time-varying discount rate q(s), we can use the transform
}{log(r)})
and produce the same problem, so long as log(q) is uniformly the same sign as log(r).
ADDED. On most subjects, I would have let my esteemed interlocutor have the last word so as to keep the peace and so as not to appear as a self-aggrandizing jerk who cannot stop trying to get one up on the person I am disagreeing with. I humbly suggest however that in subjects like math where there often is an objectively-correct fact of the matter, everyone benefits a lot from writers not being too afraid to be confrontational. One of those benefits is “clarity” (something concrete for the reader’s mind to latch onto), something easily lost in the abstractions in conversations about math. In other words, I humbly suggest that a competent writer involved in a dialog about math will appear to observers who are not used to good dialogs about math to be unnecessarily domineering, rude, dogmatic or otherwise socially inept even if he is not.
ADDED. In other words, in internet discussions on math (or programming languages), if you care too much about not insulting or embarrassing your interlocutor, my experience has been that the whole discussion tends to become a hazy fog.
ADDED. I am open to learning from others here how to improve the social side of my communications in dialogs like this.
I suspect there’s confusion over what it means to have different discount rates / utility functions at different times. This could mean either that utility depends on the time (call it τ) at which it’s computed, or on the time (call it t) at which utility-bearing events occur. The latter alone is always OK, whether or not the relationship is exponential. The former alone might create dynamic inconsistency, and if so, probably (always?) a money pump. Dependence on t-τ (i.e., ‘discounting’ as usually conceived of) is dynamically consistent if and only if the relationship is exponential.
I am open to learning from others here how to improve the social side of my communications in dialogs like this.
Given that the confusion between you and RobinZ was dispelled below, a good piece of advice might be to be careful when you think you have an interlocutor trapped between a simple theorem and a hard place; it’s often turned out (in my experience) that some condition of the theorem doesn’t apply to the particular case the other person is suggesting, and that the divergence of opinions can be traced elsewhere.
Most of the regulars here are smart enough to get the point on preference reversals when pointed out— the fact that RobinZ said he understood but was talking about something different should have counted as evidence to you.
I am almost certain that I am simply not conveying what I mean—I don’t think you’re self-aggrandizing, I think you’re as frustrated as I am with this obstinate (apparent?) disagreement.
I’m going to describe a concrete example. If you’re right, you should be able to either (a) explain how to perform a money-pump on the agent described, or (b) explain why the agent described constitutes a special case. If I’m right, you should be able to describe the difference between the agent that would suffer preference reversals and the agent described.
Let t represent the number of years since 2000 C.E. Let E(t) represent an earnings stream—between time t and time t+dt, the agent gains revenue E(t)*dt. Let r(t) represent the instantaneous discount rate at time t. And let P(E) represent the value of earnings stream E to the agent at the year 2000. (The agent is indifferent between earnings stream E and immediate revenue P.)
When r(t) = r is a constant, we can easily calculate the present value of any instantaneous future earnings dE at time t:
dP=dErt
which corresponds to the simple formula
logdPdE=−tlogr=∫t0−logrdt
I maintain that this last formula,
logdPdE=∫t0−logrdt
still holds when r is no longer a constant, and therefore (as dE = E(t)dt):
P=∫t0E(z\exp{\int_0^z-\log{r(y)}dy}dz)
Note that for the special case of F_t—future earnings at time t—we have
The OP gives two examples of market pricing—the market price for a website, and a perhaps more subjective price of acquiring a marketable skill set. The question of how to value cashflows to determine a market price has been pretty well studied. The fundamental theorem of arbitrage-free pricing basically boils down to saying that to avoid arbitrage possibilities in pricing, risk-adjusted cashflows must be discounted at a risk-free rate.
The scope of this theorem is continuously traded securities; it seems reasonable to apply inductive logic to extend this result to any commodity well modeled by a Walrasian auction. This would include, I think, a marketable skill set.
When the OP talks about ‘my discount rate’, he must be referring to his personal preferences—i.e., his utility function.
I don’t know much economics, but I think the point I was making was that other utility functions were possible. I don’t have any comment on pricing risk.
The problem with your math is that your r(t) isn’t the same as my r. If v(Deltat) gives the factor by which you discount a future resource acquisition that will occur Deltat in the future, then r is for people who have a v(Deltat) of the form
I’m not suggesting “treat r as a constant, but let it vary”—I’m suggesting “treat r as a fixed function of time t”. For example, someone might use
r(t =\frac{1}{(T-t)^2})
to value resources proportionately to how long before time T they are acquired. This kind of valuation scheme is not vulnerable to money-pumping.
Edit: My math looks wrong on the equation, but I hope the example illustrates what I mean when I say “r is not constant”.
When you say, “Someone might use r(t) = . . . to value resources . . .” and then refer to the equation “r(t) = . . .” as a “valuation scheme,” you lead me to believe that you believe that r represents utility. But that is not how John is using r. In John’s section on net present value, John has postulated that the utility experienced by our plucky entrepreneur at time t == u(t) == a / r ^ t where a and r do not vary over time.
Although John did not explicitly mention the function u, I do so now because you seem to have confused u with r.
You wrote that r can very over time “without self-contradiction”, to which I replied, “not without our plucky entrepreneur becoming a money pump,” which I still believe to be the case. Of course, John’s model does not capture the full complexity of the choices and constraints facing an entrepreneurial software developer, but there is a good reason why most treatments of net present value assume that the discount rate does not vary over time.
I did not misunderstand. The discount over a time period dt with a constant r is 1/r^dt. If we want a time-varying discount rate q(s), we can use the transform
and produce the same problem, so long as log(q) is uniformly the same sign as log(r).
I am sad because my attempt to teach you about preference reversals has almost certainly failed.
ADDED. For reference, here is the whole dialog on preference reversals.
ADDED. On most subjects, I would have let my esteemed interlocutor have the last word so as to keep the peace and so as not to appear as a self-aggrandizing jerk who cannot stop trying to get one up on the person I am disagreeing with. I humbly suggest however that in subjects like math where there often is an objectively-correct fact of the matter, everyone benefits a lot from writers not being too afraid to be confrontational. One of those benefits is “clarity” (something concrete for the reader’s mind to latch onto), something easily lost in the abstractions in conversations about math. In other words, I humbly suggest that a competent writer involved in a dialog about math will appear to observers who are not used to good dialogs about math to be unnecessarily domineering, rude, dogmatic or otherwise socially inept even if he is not.
ADDED. In other words, in internet discussions on math (or programming languages), if you care too much about not insulting or embarrassing your interlocutor, my experience has been that the whole discussion tends to become a hazy fog.
ADDED. I am open to learning from others here how to improve the social side of my communications in dialogs like this.
I suspect there’s confusion over what it means to have different discount rates / utility functions at different times. This could mean either that utility depends on the time (call it τ) at which it’s computed, or on the time (call it t) at which utility-bearing events occur. The latter alone is always OK, whether or not the relationship is exponential. The former alone might create dynamic inconsistency, and if so, probably (always?) a money pump. Dependence on t-τ (i.e., ‘discounting’ as usually conceived of) is dynamically consistent if and only if the relationship is exponential.
That agrees with my suspicions—thank you.
Given that the confusion between you and RobinZ was dispelled below, a good piece of advice might be to be careful when you think you have an interlocutor trapped between a simple theorem and a hard place; it’s often turned out (in my experience) that some condition of the theorem doesn’t apply to the particular case the other person is suggesting, and that the divergence of opinions can be traced elsewhere.
Most of the regulars here are smart enough to get the point on preference reversals when pointed out— the fact that RobinZ said he understood but was talking about something different should have counted as evidence to you.
I am almost certain that I am simply not conveying what I mean—I don’t think you’re self-aggrandizing, I think you’re as frustrated as I am with this obstinate (apparent?) disagreement.
I’m going to describe a concrete example. If you’re right, you should be able to either (a) explain how to perform a money-pump on the agent described, or (b) explain why the agent described constitutes a special case. If I’m right, you should be able to describe the difference between the agent that would suffer preference reversals and the agent described.
Let t represent the number of years since 2000 C.E. Let E(t) represent an earnings stream—between time t and time t+dt, the agent gains revenue E(t)*dt. Let r(t) represent the instantaneous discount rate at time t. And let P(E) represent the value of earnings stream E to the agent at the year 2000. (The agent is indifferent between earnings stream E and immediate revenue P.)
When r(t) = r is a constant, we can easily calculate the present value of any instantaneous future earnings dE at time t:
dP=dErt
which corresponds to the simple formula
logdPdE=−tlogr=∫t0−logrdt
I maintain that this last formula,
logdPdE=∫t0−logrdt
still holds when r is no longer a constant, and therefore (as dE = E(t)dt):
P=∫t0E(z \exp{\int_0^z-\log{r(y)}dy}dz)
Note that for the special case of F_t—future earnings at time t—we have
Sorry, the concrete example. Take
and point future income functions
which (using the Dirac delta function) correspond to instantaneous incomes at times t = 10 and 20. That is, 2010 and 2020.
Using these functions,
and
Note that to find (say) the value of F_2 in 2010, you would write
which is not equal to P(F_1).
The OP gives two examples of market pricing—the market price for a website, and a perhaps more subjective price of acquiring a marketable skill set. The question of how to value cashflows to determine a market price has been pretty well studied. The fundamental theorem of arbitrage-free pricing basically boils down to saying that to avoid arbitrage possibilities in pricing, risk-adjusted cashflows must be discounted at a risk-free rate.
The scope of this theorem is continuously traded securities; it seems reasonable to apply inductive logic to extend this result to any commodity well modeled by a Walrasian auction. This would include, I think, a marketable skill set.
When the OP talks about ‘my discount rate’, he must be referring to his personal preferences—i.e., his utility function.
I don’t know much economics, but I think the point I was making was that other utility functions were possible. I don’t have any comment on pricing risk.
The problem with your math is that your r(t) isn’t the same as my r. If v(Deltat ) gives the factor by which you discount a future resource acquisition that will occur Deltat in the future, then r is for people who have a v(Deltat ) of the form
v(Deltat = \frac{1}{r}^{\Delta%20t})
Your equation is for people who have a
where k is some constant.
vDelta