Opportunity Cost (Ferraro-Taylor): how partial equilibrium story can go wrong

"You won a free ticket to see an Eric Clapton concert (which has no resale value). Bob Dylan is performing on the same night and is your next-best alternative activity. Tickets to see Dylan cost $40. On any given day, you would be willing to pay up to $50 to see Dylan. Assume there are no other costs of seeing either performer. Based on this information, what is the opportunity cost of seeing Eric Clapton? (a) $0, (b) $10, (c) $40, or (d) $50."

 The above is the question asked at the 2005 annual meeting of AEA. For the news link, see http://www.nytimes.com/2005/09/01/business/the-opportunity-cost-of-economics-education.html .

 The official answer given by Ferraro-Taylor is (b). The reason goes as follows. Think of standard partial equilibrium story (for the people unfamiliar with the term "partial equilibrium": it's equilibrium of single market - simple equilibrium story you probably already know.). In introductory economics classes (or called principles classes, or econ 101 classes), you learn the concept of economic surplus that includes consumer surplus. In this example, this individual's consumer surplus by purchasing Dylan ticket is $10, since the agent was willing to pay at maximum $50 but now got the ticket by $40, and the number of the ticket you buy is 1.

 $10 is the (net benefit) value of the Dylan ticket. Thus this is the opportunity cost of seeing Eric Clapton, because opportunity cost is defined as the value of the best alternative forgone. (but see below for another common definition)

 So far everything makes sense. And it is, and those economists under job market stress were dumb. Except, I would like to think of this question from a different light.

 Think of a typical derivation of demand curve/function in introductory economics classes (partial equilibrium). There are two goods \(X\) and \(Y\), with \(X\) being an "aggregated good" of all other types of goods that are not \(Y\), assumed to be infinitely divisible. \(Y\) here is either the Clapton concert or Dylan performance. Let \(Y\) be the Dylan performance ticket, which is labelled as \(Y_d\). The agent's utility function is \(U(X,Y_d)\), with budget constraint \(P_x\cdot X + P_{y_d} \cdot Y_d = W\), where \(P_x\) refers to price of \(X\) and \(P_{y_d}\) refers to price of \(Y_d\). In the partial equilibrium story, \(P_x\) is assumed to be already determined and given. Thus what varies is \(P_{y_d}\), and we think of how \(X\) and \(Y_d\) vary as \(P_{y_d}\) changes. Assume for now that total budget \(W\) is fixed.

 In the given question, \(Y_d=0\) or \(Y_d=1\). Thus, assuming no saving is done, if \(Y_d=0\), then \(P_x \cdot X = W\). If \(Y_d=1\), \(P_x \cdot X + P_{y_d} = W\). Now it is said that the agent is willing to pay maximum of $50 for the Dylan ticket.  That suggests \(U([W-50]/P_x,Y_d=1) \geq U(W/P_x,Y_d=0)\) (or if one insists on continuous utility, equality, but this does not matter).

 Now think of \(U(X,Y_c)\), where \( Y_c\) is the Clapton ticket. Budget constraint for combination of \(X\) and \(Y_c\) is \(P_x \cdot X = W\), as the Clapton ticket is free. Thus, utility function is \(U(W/P_x,Y_c=0)\) or \(U(W/P_x,Y_c=1)\). Assume that \(U(W/P_x,Y_c=1) > U(W/P_x,Y_c=0)\).

 Now one sees a different picture. Opportunity cost of the Clapton ticket being $10 is supposedly said to mean that the agent must be willing to pay at least $10 for the Clapton ticket if the agent is to go to the Clapton concert. But is this true? Translating this into utility, this suggests that \(U([W-10]/P_x,Y_c=1) \geq U([W-40]/P_x,Y_d=1)\). Assume additively separable utility.
 The aim is to violate:
$$U([W-10]/P_x,Y_c=1) \geq U([W-40]/P_x,Y_d=1)$$
within the following utility specifications/constraints:
$$U(X,Y_c=1) = V(X) + K_c$$ $$U(X,Y_c=0) = V(X)$$ $$U(X,Y_d=1) = V(X) + K_d$$ $$U(X,Y_d=0) = V(X)$$ $$U([W-50]/P_x,Y_d=1) \geq U(W/P_x,Y_d=0)$$ $$U(W/P_x,Y_c=1) \geq U([W-40]/P_x,Y_d=1)$$
 The main inequalities that we wish to satisfy then are numbered #1, #2:
$$U([W-10]/P_x,Y_c=1) < U([W-40]/P_x,Y_d=1) \leq U(W/P_x,Y_c=1)$$ $$U([W-50]/P_x,Y_d=1) \geq U(W/P_x,Y_d=0)$$
 Suppose \(W=210\), \(P_x = 40\), \(V(x) = \sqrt{x}\). Solving the #1 first, one gets:
$$\frac{\sqrt{17}}{2} - \frac{\sqrt{21}}{2}\leq K_c - K_d < \frac{\sqrt{17}}{2} - \sqrt{5}$$
 This inequality is consistent, meaning that there exists a pair \((K_c,K_d)\) such that the inequality is satisfied. #2 only governs how \(K_d\) must be set, and thus once appropriate \(K_d\) is obtained, one get use #1 to get appropriate \(K_c\).
 One may say #2 must be equality, but it is easy to see with the example that this does not matter.
 Thus, "the agent must be willing to pay at least $10 for the Clapton ticket if the agent is to go to the Clapton concert" is false in this consideration.
 This raises the question: did we in fact miscalculate opportunity cost? In fact we did. Because in this consideration, opportunity cost should have been the net benefit value of obtaining \(X\) and \(Y_d\) together. We should do comparison jointly: \((X,Y_c)\) versus \((X,Y_d)\).

 Lastly, one may say that $50 is actually the maximum price the agent is willing to pay for \(Y_d\) given comparison with \(Y_c\). But this is ruled out by "on any given day" in the question. That is, $50 is the maximum price the agent is willing to pay if \(Y_c\) did not exist.
------------------------------------

 Now one may complain that the definition of opportunity cost is not what some people are using. (I was following original Ferraro-taylor explanation.) So let us consider the alternative common definition. In this definition, opportunity cost is sum of explicit and implicit cost of some action. Explicit cost can be said as visible money costs, while implicit cost relates to benefit you may have gotten if you chose the best alternative minus the costs that are solely due to the alternative action. (or simply net benefit of the alternative.)

 In this question, explicit cost of the Clapton ticket is zero. Thus, opportunity cost is reduced to implicit cost. (one of available Internet notes for this explanation: http://faculty.arts.ubc.ca/slemche/academic_w09/webf_w09/Opportunity%20Cost%202_s07.pdf )
(to follow notation of the link, \(OC(Y_c) = DOC(Y_c) + B(Y_d) - DOC(Y_d) = 0+50-40 = 10\), but see the conclusion below)

 To summarize, the point made in this post is that while $10 is the correct answer for possibly introductory econ exams, this is really not a good answer, once we really consider what assumptions are assumed for this result. Choice of \(X\) over \(Y\) must also be considered if we are really to do proper analysis. After all, suppose that we do not consider \(X\) and only consider \(Y\). Budget is W, and assuming the agent does not save, the agent only has \(Y_c\) or \(Y_d\) for expenditure. Thus there is no point of discussing net benefit value, as for the agent money not spent is anyway useless. And one cannot use standard demand analysis here.

------------------------------------

I've been sick for three months, and I plan to write posts more frequently as I come to recover.

------------------------------------

Update (Oct 14, 2016):

 The common and most popular interpretation of the question considers $50 as the maximum price the agent is willing to pay that does not take into account \(Y_c\). That is, as the UBC (Canada) link above says, this is simply reservation price, and reservation price simply equals gross benefit/value. That is, the agent obtains zero net benefit if the good is priced at the agent's reservation price. This is the interpretation I also use to demonstrate problematic aspects of this question.
 However, the following interpretation can also be made: since $50 is the maximum price the agent is willing to pay for the day the Clapton concert is being held, this choice means \(Y_d\) is always preferred over \(Y_c\) whenever the ticket cost of \(Y_d\) is less than or equal to $50.
 Since price of \(Y_d\) is $40, this means that the agent will go to the Dylan performance instead of the Clapton concert. The opportunity cost of the Clapton concert still remains at $10, though. The alternative interpretation avoids the problem noticed in this post. 

Paul Romer's mathiness paper: a year later (1)

For this first post of the (recalling) Paul Romer mathiness series, Appendix B of https://paulromer.net/wp-content/uploads/2015/05/Mathiness-Appendix-Expanded.pdf is the main topic.
(Paul Romer blog post: https://paulromer.net/mathiness/)

tl;dr version: Paul Romer is right, but Lucas and Moll aren't really wrong also. But things to think about are there.

The paper came to my mind, as I was playing with my brain spare time capacity, while I was (and still partly am) suffering from feeling sick. (not a good way to live a life - it's not that I am like this for other times, but sometimes, you are struck with some trains of thoughts for no reason) More specifically, I was thinking of use of double limits, then I was reminded of the paper.

Since there'll be people not wanting to go alternating between the paper and this post, I will write down some of Romer's core arguments using Romer's notation. \(\beta\) is the innovation arrival frequency. Now consider the following results:
$$\lim_{t \rightarrow \infty} g_{\beta}(t) = 2\%$$
$$\lim_{\beta \rightarrow 0} g_{\beta}(T) = 0$$
\(g_{\beta}(t)\) is the economy growth rate given \(\beta\) at time \(t\). The second result holds for any \(t=T\). In this circumstance, which one of the above two equations is the right way to think of long-run growth rate? For more clarity, change the above two equations to:
$$\lim_{\beta \to 0}\lim_{t \rightarrow \infty} g_{\beta}(t) = 2\%$$
$$\lim_{t \to \infty}\lim_{\beta \rightarrow 0} g_{\beta}(t) = 0$$
The first equation choice is what Bob Lucas and Benjamin Moll used in their paper. And without (and possibly even after) reading Paul Romer's paper and appendix, many would feel that this is a justifiable choice. The second limit says: suppose that innovation arrival rate is close to zero. Then let's see what happens as one goes long-run. The first limit says: let's first move forward to long-run and see what happens.
As long as \(\beta\) isn't close to zero, the first limit is the sensible one, right?
But one can think differently. After all, it is generally assumed that we cannot really "see" infinite time, and all times are finite. If the so-called long-run growth rate is not achieved - and actual growth rate remains below the so-called long-run growth rate (the first limit one) - after a short period of time, but rather requires so much time, possibly "almost never" (because \(\beta\) is so close to zero), what is the point of talking about growth rate? Growth rate is about growth happening over time, and if long-run growth rate is not what prevails for so much of time, any conclusion following from the first limit long-run growth wiould be mathematically correct, but economics-wise meaningless.
Thus, one is forced to discuss actual economic growth paths reaching up to the first limit long-run growth rate rather than just abstracting away these processes and buidling up conclusions from the first limit long-run growth rate, says Romer or Romer per my understanding.

(side question: while the first limit long-run growth rate is "unobservable", would Romer's statements about this unobservability defeating the theory really be justified? [talking about the appendix here too.])

Now reminded of this paper, I actually was reminded of my own thought a year (I believe?) ago regarding limit. It is not exactly about the double limit issue above, but quite a similar one. And I plan to write about this after this series.