constraints

Every economics student learns the lagrange method of solving constrained optimization problems. Beyond the technicalities, the lesson is clear, if not obvious — When u are constrained, it stops you from making choices you’d made if you were unconstrained. Hence a constraint constrains you, and the math help figure out by how much. I was just reading a textbook description of a model that prohibits the return on capital to rise above a certain level, say s. There are two, and only two, possibilities: either it doesn’t matter because s is too high to constrain, or it matters greatly, and the capital decision is made to meet this constraint.

Optimal Stopping

When do you know when to quit? When to buy? This is a hard question, so as first pass, we make some assumptions.

Money has value R that you discount at rate r. The cost of the good is C, and the cost falls at rate (r+w). This is a process innovation that makes producing the good cheaper over time, and, we assume that this savings is passed on to consumers. Clearly, there is a benefit to waiting because cost is falling to zero. A standard method for discounting is exponential discounting, which gives the result that cost is falling faster earlier in time, and slower later in time. Last, the good itself has value S, which the person discounts at rate r, but enjoys from time T to forever (we assume infinitely durable goods for convenience — it won’t affect the conclusions later).

The problem is to decide the value of T, i.e. when to exchange money for the good.

Note that it must be true that S>R, otherwise, the value of the good will forever be less than the value of money. You end up never buying, which means T is infinite.

What is the basic trade-off? At the best T, it must be that waiting longer isn’t worth it. In other words, the benefit of waiting (lower cost) is less than the cost of waiting (which is foregone benefits of owning the good, which is a function of S-R). It must also be true that buying sooner won’t help, coz the cost of buying early (high cost) is far greater than the benefit of owning the good.

So, suppose we decide the value of the best/optimal T according to the rules above. What then is is the effect of a faster rate of cost reduction (a higher w)? This should push you to wait; BUT only when the cost reduction benefits of waiting are huge, which as we’ve determined is when the optimal is small. The process improvement forces you to buy earlier if the optimal T is large.

What helps determine whether optimal T is large or small? The size of (S-R). The larger S-R is, the larger the benefit of buying sooner, coz the gain you get from buying, S, is larger than the value of money, R. The larger S-R is , the lower the optimal T — in this situation, a cost improvement leads to waiting longer.

Finally, lets say there is no secular price declines for this good, w is zero. What then? Pretend you are at time zero, deciding whether to buy. If you delay, you earn a net of R-S+rC, or R-(S-rC). If R>S-rC, you think, if i delay, i get R (money happiness), which is greater than S (good happiness) minus rC (one time cost of buying the good). The opposite holds when R<S-rC.

Blast from the Past — Quasi-linear and Slutksy

Quasilinear Utility. Its linear, but not really. The numeraire good can be shown to sweep up all income effects by looking at Lagrangian Multiplier. What this means is that all additional income goes directly to buying more numeriare good. The rest of the goods’ demand are a function only of price. This has implications on indirect utility. The gain to consumers to non-numeraire good is the Consumer Surplus. So it stands to reason that changes in non-numeraire prices’ effects can be measured via Consumer Surplus. When it comes to welfare, and if we restrict our attention to changes in prices of non-numeraire, we can CV=EV=p*change(CS). Why is there a price in the last term? CV and EV are in money units, while CS is in terms of real numeraire goods. We need to convert it to money.

Lets translate this into regular terms. A numeraire good is all other goods, while the quasilinear good is say ballpens. A marginal increase in wealth leads to it going directly to all other goods. The gain in ballpen consumption is measured by CS, or WTP-price, summed over quantity bought. Lets say the price of ballpens fall. You consume more pens, and the gap between Willingness to Pay and the price grows even more. This is win for consumers, unambiguously. For the consumer, this ‘ballpen windfall’ has a conversion rate in terms of the numeraire. For example it could be ” for 100 more ballpens this year, for me, that would be equivalent to 5 units of a composite of everything else”. We make that translation, and multiply it by the price composite/index of everything else, to measure it in money. Viola! — the money value of 100 more ballpens.

Next, the Slutsky Equation. What happens to demand when the price rises? Two distinct effects. The substitution effect leads you to consumer less of the relatively more expensive good. The other effect is the wealth effect. Typicaly, higher prices lead to lower real wealth, which lowers quantity demanded too. But this is not the only case. Suppose that alot of your income comes from your selling of this good. When the price rises, you become richer, which causes you to buy more of everything, including the relatively more expensive good.

Classic example is leisure, and the price of leisure is wage. Higher wages lead to higher incomes, which encourage you to ‘buy’ leisure by not working. This leads to the backward bending supply curves. This effect is stronger the larger the weight of labor income in total income. Why? An increase in  wages increase total wages by 10% if all your income is labor income. But it increases it by only 1% if wages is just 10% of total income.

failure is great

I read an article today on happiness. A french psychologist wrote that one of the common factors linking unhappy people is an excessive concern about the past and/or the future. Not that the future or the past is unimportant, but that happy people are bothered less by the uncertainty of future outcomes and the sunk nature of the past.

A happy person must be in the ‘now’, and actively experience ‘now’, instead of constantly thinking about the future or the past.

One can see a relationship between happiness and other variables like religion. Religion helps us manage the uncertainties of the future (i.e. reaching heaven is the most important thing), and our past mistakes (God will forgive us). This is also potentially the reason why income is positively related to happiness– it is the psychological security of money.Of course, in rich countries, this effect is largely negated by a consumerist culture that demands more and more things, which by and large creates more anxiety about the future.

Optimal product launch strategy

There is a paper in our seminar schedule about the monopolist’s product launch strategy in multiple markets. Its highly stylized, but heck, its a start.There are N > 1 markets in which the monopolist can launch her product. The monopolists decision is to decide how many markets in which to launch. She can launch in all, in some, or none of these markets. For some reason, (unexplained, but i assume for simplicity), price p is fixed, at say 1/2. The good is either high quality, and its value v is 1, or low quality, where in its v is 0. The monopolist doesn’t know the quality of the good, which explains why we have non-discriminatory pricing. If the monopolist knew about her products quality, she would set two different prices and there would be a separating equilibrium. [Which raises an interesting question: how can a monopolist not know the quality of the product?]

The consumers belief about the value (lambda) takes values from (0,1). The consumer buys v-p>0 [This is confusing, it should be lambda-p>0? Is lambda the distribution of consumer's unit demands? I think the author wants to say that lambda is the same for each consumer in a market, which means its just like assuming one consumer per market, not terribly interesting].

Whats interesting is the updating method. Imagine you are in a market, and you observe a ’signal’ (she doesn’t explain what this signal might be, but this is a random signal) from another market where the product is launched. This signal can either be good or bad. A good signal may mean that the value of the good is 1 (the highest possible). A bad signal is defined similarly. A signal is independently and identically distributed across markets and time.

However, like many signals in life, signals are noisy. Let q = Prob(signal=1|v=1) =Prob(signal=0|v=0). That is, q is the probability that the signal is correct.We assume here that signals are more likely to be correct than not: q>1-q. [note: if this isn't true, then the graph i present below will switch].

The updating of beliefs on the product’s value is of interest to the monopolist. The monopolist would prefer to launch sequentially, IF she can manipulate willingness to pay in markets where she has not launched yet. In terms of the signal, if the signal is good, this implies, by Bayesian updating, those who see this signal would revise their beliefs/lambdas upward which raises demand.

Put in simply, this is kinda like viral marketing. Launch a product for a small group first. They tell the world that the product is good once they consume it and know its good. The rest of the world sees the signal and says, hey, windows vista is better than i thought… so they revise their valuations upward based on this signal. Similarly, if the signal is bad, they revise their values downward.

A consumer in a market where the good has not been launched has two pieces of information. The performance in the first market(s) in which it has been launched, as well as the random signal defined earlier.

Below you find a graph depicting the bayesian updating of beliefs:

Given that the price is 1/2, if lambda=lambdalow or less, then the succeeding markets will never buy the good, no matter what the realization of the signal. Hence you will never launch the good. We say that the consumer here is extremely pessimistic.

If lambdalow<lambda<p=1/2, then something interesting happens. Here, we say the consumer is pessimistic — the prior belief is that the good is ‘overpriced’.

Assume for simplicity two markets, and we consider a sequential launch which simply means launching one market at a time. Lets say the good was succesful (which means it received a good signal) in the first market; this leads to an updating upwards for the second market. But there is also a random signal A good random signal leads people to buy, a bad random signal lowers their lambda further. So,if the signal in market one is good and the signal is market two is good, there is no difference between sequential and simultaneous launch. Both markets buy anyway. If the signal in market one is good, and the random private signal in market 2 is bad, then the two signals cancel each other out, leading to lower total sales (good sales in market 1, no sales in market 2) — an inferior outcome which is came from the fact that the product was launched sequntially. If the signal if the first market is bad, and the second is bad, no one buys — but this would be true in a simultaneous launch too.

So the dominant strategy is to simultaneously launch.

[Note: its important here that you have representative, identical consumers in each market to get this result. If you abandon these, the analysis becomes more involved, but still follows the general line of reasoning outlined above]

If p=1/2>lambda<lambda high, something interesting happens. This time, the consumer is optimistic. Here, if you have simultaneous launch, each market will receive a random signal of s, which causes some to buy, others not.

If the launch is sequential, there are two signals for the later market: success/failure in the first and a private random signal in the second market.

The reasoning is similar to the preceeding case. If the signals are opposing, then they cancel out — but thats fine because the lambda>p and they (the second market) still buys. If the signals are the same, clearly both markets buy the good. We see why sequential benefits the monopolist — its a form of random signal hedging. A sequential launch allows opposing signals to cancel outwhich leads to higher total sales, while in a simultaneous launch, opposing signals mean random successes which is inferior.

Finally, if lambda>lambdahigh>1/2, then no matter what the realization of the signal, each market buys the product. so sequential is the same as simultaneous.

This is nice model that allows us to see how sequential launches help a monopolist when there is information assymmetry both for the seller and the buyer. They way the author explains the mechanism and demand was confusing at first. Also, the next step would be to endogenize the signal as a function of market variables, like sales and quantity and price. I also don’t like that latter markets have two signals. I’d rather model it as one signal, and then each market is heterogenous along a certain dimension. In this way, we might be able to get the result that if a sequential launch is implemented, then the first market must have the larger market size, or higher profit potential. This is much more realistic.

Where is the Love?

This article from Time wants to explain the disappearance of the love story. There are demand side issues (We’re sick of love stories!) and supply side innovation issues (I can’t make original love stories anymore!). It concludes with a comment on the structure of the industry. Titanic, the most successful movie in movie history, was a slow starter and needed time to earn its billion dollar take. Because of competition, supply side costs, its more important than ever for films to make alot of money on its first weekend. This means you have to attract people who watch movies on their opening weekend — young men.

Confused About the Persistence of Poverty

I received an email recently about the book ‘Persistence of Poverty’. The email was about blogosphere reaction to the book, particularly to the implication on the dimishing marginal utility:

LIKE Tyler Cowen and Megan McCardle, I’m intrigued by the thesis of Charles Karelis’ new book “The Persistence of Poverty”, which was discussed Wednsday by Steven Pearlstein in the Washington Post. Pearlstein describes the kernel of the book thus:
But what if [the law of diminishing marginal utility] is wrong? What if it doesn’t apply at every point along the income scale? If you and everyone around you are desperately poor, maybe it’s perfectly rational to think that an extra dollar or two won’t make much of a difference in reducing your misery. Or that you won’t be able to “study” your way out of the ghetto. Or that if you find a $100 bill on the street, maybe it’s logical to blow it on one great night on the town rather than portion it out a dollar a day for 100 days.

i find tyler cowen and Megan McArdle’s praise strange. i need to review the finer points of empirical microtheory, but i suspect diminishing marginal utility is something you cannot prove.

In data, you don’t observe utility. You observe choice. Assuming rationality, choice is a function of two things: preference and the budget set. When we observe the poor not doing what a ‘theory’ says they should do, any of these three things can be the culprit.

Consider a chapter in our grad text book in macro, Advanced Macroeconomics by Paul Romer — Consumption and the Lifecycle hypothesis. It starts with a model, with assumptions more restrictive than above (time separable preferences). Then there is a prediction on consumption, some of which is not borne out by data. At this point, you can abandon any of the assumptions, but most of the changes (not all, however) are changes that have a chance for empirical confirmation. Changes in the shape of the marginal utility is not one of them. Debt constraints would be a better candidate.

The Opportunity Cost of Teeth

Rose Byrne is a gifted actress, and as the photograph suggests, extremely beautiful. I may be biased, but i’m not alone. In an interview, she says she gets typecast as a victim because of her droopy eyes. If producers are typecasting her thusly, its unfortunate; i’ve seen her in comedies and dramas, and her droopy eyes can communicate whatever it is that’s required by the role.

While she’s almost perfect, there is one physical problem with her face. In this case, its her teeth. I’ve noticed recently that one of her lower teeth is misaligned. A casual observer wouldn’t notice this at all, as the lower lip mostly obscures it even when she smiles.

At first blush, this is a puzzle. Rose’s looks are her stock and trade. As an actress, she has all the incentive and the means to fix her lower teeth alignment problem. I consulted a friend whose knowledge of braces is rivaled only by her love of pleather dogs. Its clear that the cure would be worse than the disease; braces would be invasive, jarring and obvious.

This would be the kiss of death for a working actress.

My teeth-sage of a friend also led me to discover a new product Invisalign. This seems to be the product for her. I’m sure she’s aware of it, and maybe over time we’ll see a change if you know where to look.

As a fan, I’m a firm believer that imperfection makes a beautiful woman even more breathtaking. I’m happy her lower teeth are slightly askew. It makes her more real to me, as i have mis-aligned teeth as well.

Advertising and the Rational Man

In the AMC series “Mad Men”, Don Draper, an ace Ad-exec at the fictional (but no doubt prestigious) NY advertising firm Sterling-Cooper was sharing a cocktail with potential client, and possible lover, Rachel Menken. He learns that she has never married because she has never been in love. Draper leans back, and in a world-weary voice says,

“The reason you haven’t felt it is because it doesn’t exist. What you call love was invented by guys like me to sell nylons.”

A tad exaggerated perhaps, but it has a sliver of truth. Advertising is one of the most ubiquitous forms of mass media, and its very prevalence irritates economists.

Consumer theory can be distilled to two simple questions, plus two critical assumptions.

a) What are my feasible choices?
b) What is my best choice, considering my set of feasible choices?

Since almost any choice in a feasible set can be justified, an additional assumption is needed, which is called revealed preference or consistency. If two objects are in a feasible set, imaginatively called A and B, and A is chosen over B, then whenever A and B are both feasible, A will always be chosen over B. A good way to put it is the feasible set and the individuals preferences are independent of each other. The second assumption is that the individual doesn’t suffer from a lack of appropriate knowledge: i.e. he knows the problem, he is aware of the choices, he knows what he wants, etc…

Of course there are many technical details that need not interrupt the rest of this post. These juicy bits form many a grad school midterm, final and prelim, and are surely not the victims of neglect. The important thing to note is that this is the economic model of the Rational Man.

Many people believe that this is not a literal description of human behavior. In fact, economists don’t believe in the literal interpretation. Like sophisticated exegistes, they believe in the model in the sense of an intelligent lay-person believes that the bible isn’t the literal truth but is the correct interpretation of God’s will.

Specifically, even if people don’t make decisions like the rational man, people behave as if they do. It is people’s actual behavior that interests economists, not the exact procedure that generates this behavior.

Thus, ANY model of behavior that generates consistent choices (briefly defined above as consistency). This is the reason why assumption two is a non-starter – economists don’t care if people can’t compute with cray-like ability. Ultimately, as long as people’s behavior fits the rational man model, whether or not man makes choices in this particular way is irrelevant.

However, the key problem with this paradigm is this: what if the choice set and preferences are NOT independent? In this case, the procedure of the choice is important. To make this as stark as possible, suppose that A, B and C are elements in a set, and given this our individual wants A. However, if ABCD are the elements, B is the choice.

You may be asking: how relevant is this? This phenomenon permeates human decision making. You need only look at the psychology literature. In his book “Modeling Bounded Rationality” by Ariel Rubinstein, he summarizes the issue as the following:

a) Framing Effects
b) Tendency to Simplify Problems
c) Search for Reasons

These effects do indicate that advertising makes sense from an economic point of view. The consequence of the fact that preferences are changeable in very systematic ways is that purchasing decisions can be affected by marketing and advertising. One final rationalization of the rational man approach is that the three effects are mistakes, in the sense that once they know it’s a mistake, consumers or traders change their behavior accordingly. Merely labeling them as mistakes is not useful because the fact that they are systematic makes them economically relevant.

The important big-picture questions then are: the extent to which preferences or procedures are manipulable, does this persist, and how do people learn about their ‘mistakes’.

Looking at the nitty-gritty details, Prof. Rubenstein explains how to model these three effects, which I’ll detail later. This is a technical challenge, and is a non-trivial change in the rational man model.