Follow Up On Measuring Utility

Jonathan came back with this:

Hi, thanks for that Professor. IMO Utility does describe the feeling, or at least the “something” that kicks-in mentally, particularly with ""money"" based decisions - when you know (or don’t) you are detaching yourself from the wholly rational course of action (expected return) to take the money on the table offered, or a sum offered with certainty that is just enough of a push - for you to cash in your chips.
Where I’m stuck – is that there is a load of books and research about the abstract pro’s and cons – but I can’t find anything anywhere on the net that explains just how to go about discovering how to assess yourself (e.g. a clear example from the grass up).
Any ideas?
Thanks again,
Jon   

My response:

If you took the standard expected utility theory at face value, then you would approximate the utility function locally with a quadratic.  The benefit of doing that is you get that for small gambles around the mean, the theory says the risk premium should be the Arrow-Pratt measure of absolute risk aversion at the mean, r, times the variance of the gamble.

For example suppose you face the gamble of $1,001 with probability .5 and  $999 with probability .5, so the mean is $1,000 and the variance of this gamble is $1, which is small relative to the mean.  You then try to elicit what amount of money for certain would make the person indifferent between having that or having the gamble.   Suppose you find the certainty equivalent determined experimentally is $999.60.  So in this case the risk premium is $.40 and hence the inferred Arrow-Pratt measure of absolute risk aversion is .4.

Now if you do this seriously, you would like to see whether the theory is really confirmed.  So you might try other small gambles with mean $1000.  For example you might consider the gamble (a) of $1001 with probability .8 and $996 with probability .2  as well as the gamble (b) of  $1004 with probability .2 and $999 with probability .8.  Each of these gambles has the same variance, $4.  So ahead of time you might guess based on what you discovered before that the measured risk premium would be $1.60, which must be the case if  the formula in the first paragraph held exactly and you measured the risk premium perfectly in the previous experiment.  You might get something close to that for the gamble (a) but you definitely won't for gamble (b) because that says the certainty equivalent is $998.40, which is lower than the $999, what is attained in the lower income state.

There are two possible source of error here: (1) measurement of the risk premium in the first experiment and (2) the formula that relates risk premium to the Arrow-Pratt measure and the variance of the gamble.   The second error becomes less as the variance gets smaller but the first error gets bigger that way.  So even if you take the theory as fully correct, you will have issues in measuring the utility function.

Let me make one more point on this.  The psychologist Daniel Kahneman, winner of the Nobel Prize in  Economics, has shown that the standard expected utility theory is wrong and that something else called Prospect Theory is closer to how we actually behave.  In that a reference point matters for evaluating gambles and then whether the outcome is a win with respect to the reference point (where the individual is then risk averse) or a loss with respect to the reference point (where the individual is then risk seeking).  Put another way, the utility function for Prospect Theory is convex-concave, with an inflection point at the reference point.   If you find this interesting you might read Kahneman's recent book, Thinking Fast and Slow.

Can (Expected) Utility Functions Be Measured Empirically?

Jonathan asked:

Hi Professor Arvan,
I just watched your ExpUty video on Youtube -
In reality, how would you go about capturing personal utility functions and preferences? Is there a defacto approach / way or template for doing this for Money, or other goods? I referring to the question construction, interpretation / ranking of the answers and then the maths behind plotting the curve? Or do you know of a spreadsheet / program solution? I take it ""Utils"" can only ever be ordinal, in reality?  I would appreciate any further advice on the subject - Thanks, Jon

My response:

There are lots of issues that question.  So it is a good one in bringing those to the surface.  Let's get to some of these:

(1)  Is the person rational a la the expected utility hypothesis or do "animal spirits" better serve as a guide to behavior?  And here instead of animal spirits think of Darwin and the decision to fight or flee. Moderate financial risk is qualitatively different, in my view, than the threat of somebody doing physical violence on your person, or the chance you may catch some serious disease.  For the latter two, I doubt expected utility theory is useful at all.  For the first, at least there is some hope it might be.  

(2)  How does the person assess the probability distribution in practice?  We understand how to do this in coin flipping, or casino games, but for real-world uncertainty do probability assessments at all conform with what the actuaries tell us we should believe?  There is psychological research on this and it confirms that people are bad at making probability assessments on their own and typically over estimate the chance that a threat will materialize.  The expression is "better safe than sorry" and the research supports that conclusion.  But it also means the individual is not being rational in the expected utility sense.  On the flip slide of this people of modest income are known to buy lottery tickets, even when the odds are quite bad for them.  They are fascinated with the prospect of a high payoff, irrespective of the odds.

(3) When there is more than just one good, money, but rather several commodities does it make sense to monetize them all and speak of a single dimension of risk preference or is it harder than that?  As far as I know there is no good theory of risk preference in a multi-dimensional commodity setting.  Since consumption bundles are themselves random - for example, if you buy a knock off computer instead of a name brand to save a few bucks how well does it function - the issues certainly appear there but whether there can be a coherent risk preference theoretically, I doubt it.  I do think that psychologically we tend to convert these sorts of risk into unto time units - as a measure of the possible inconvenience - and if necessary then try to monetize those, but we do it only in a very rough way.

(4)  Are a person's risk preferences stable over time or do they vary?  Let me give just one example here.  People may drink alcohol because it "loosens them up," which you might interpret as becoming less risk averse.  If the choice to drink alcohol in the first place is rational, and some might question that, then it is as if the risk aversion is a constraint that the person wants to shed.  (And this is why there is so much discussion about peer pressure and drinking, because it may be others who want the person to shed the risk aversion, not the person himself or herself.)  There are certain circumstances  where a normally mild person (one who will take flight most of the time) becomes extremely aggressive (opts to fight and then does so with a fiery intensity) so it's almost a Dr. Jekyll and Mr. Hyde thing.

Conclusion.  Given these various caveats, each which bring realism to the story, you might ask whether expected utility is at all useful as an approach.  I would say, yes it is useful especially if you restrict the domains where you apply it.  The first is that it provides a nice explanation of the demand for insurance.  The second is that in trading risks across individuals, it offers the reasonable intuition that with increasing wealth risk aversion should decline simply because there are better opportunities for diversifying the portfolio as one gets wealthier and hence suggests where there may be gains from trade from better sharing risks.

Perfect Complements Again

Norman asked:

We have a Leontief utility function like U(x1,x2)= min (x1,x2).What i want to do is to solve x1 and x2.  I have read a lot about the theory but i couldn't find any solved example for Leontief utility maximization. I really will be very pleased if you can give clues for that problem. Thank you for your treasure time.

My response:

Take a look at this post and see if that does it for you.  If there are further questions post them as comments here.

Consumer Surplus

Serdar asked:

For the arguement  of "Compansated variation is always bigger than consumer surplus under all price changes", could you please discuss whether it is true or not by drawing the necessary graphs? And i would be pleased if you can give a numerical example to support the arguement (utility function is a Cobb-Douglas utility function). thank you in advance.

My response:

This is discussed in the video, CV EV and Change in CS.  The graph below is from the spreadsheet used to make that video.  Let's review the definitions of CV and CS and then consider the determinants of which is bigger.

CV - this is the area to the left of the compensated (Hicksian) demand curve for the original optimum between the original price and the new price.

Decrease in CS - this is area to the left of the ordinary demand curve between the original price and the new price.

Remember that the compensated demand measures the substitution effect only, but that the ordinary demand measures the substitution effect and the income effect in combination.  For a good where there is no income effect, CV = Decrease in CS.

More generally, what matters are:

(1) the direction of the price change, and

(2) whether the good is normal or inferior.

In the graph above the original price is given by the height of the dashed horizontal line.  Then the price rises and the new price is indicated by the height of the dotted horizontal line.  The blue curve is the ordinary demand curve.  The red curve is the compensated demand curve for the original optimum.  In this diagram, the blue curve is more elastic at the original price than the red curve.  That will be the case for a normal good.  The area to the left of the red curve between the two prices is greater than the area to the left of the blue curve between the two prices. Thus in this case CV > Decrease in CS.  

I leave it to you to consider the case of a price decrease and/or the case where the good is inferior.  By the way, if the utility function is Cobb-Douglas, then the good is normal.

Isoquant and Isocost

A student posted the following question:

"The owner of a small car-rental service is trying to decide on the appropriate numbers of vehicles and mechanics to use in the business for the current level of operations. He recognizes that his choice represents a trade-off between the two resources. His past experience indicates that this trade-off is as follows:
Vehicles:     100  70  50  40  35  32
Mechanics:  2.5  5   10  15   25   35
a] Assume that annual (leasing) cost per vehicle is $6,000 and the annual salary per mechanic is $25,000. What combination of vehicles and mechanics should he employ? - Already got the answer to which is 70 vehicles and 5 mechanics
b] Illustrate this problem with the use of an isoquant/isocost diagram. Indicate graphically the optimal combination of resources.

Thank you for your help and time"

My response:

Your goal here should be not just to answer the question but more importantly to gain some understanding for why the answer is correct.  To do that you should solve this both numerically and graphically.

Numerically - The various combinations of points given lie on the same isoquant.  There is economics in computing the slope of the segment between consecutive points.  The absolute value of that slope is called the rate of technical substitution.  (Sometimes the word marginal precedes that so the entire expression is marginal rate of technical substitution.)  You should compute the full schedule of RTS.

Then you should compute the ratio of the input prices.  Students doing this for the first time are unsure whether that ratio should be (price vehicles)/(price mechanics) or (price mechanics)/(price vehicles).  The graphical approach should help there.  Which input is on the x axis?  the y axis?  Knowing that you can plot a line of constant expenditure on input bundles.  The input price ratio you want is the absolute value of the slope of the line.

The economics is in understanding when the RTS does not equal the relative price:  movement in which direction along the isoquant will result in lower cost?

Graphically -  You should plot the isoquant.  In the same diagram you should plot several isocost lines including one that lies entirely below the isoquant and another that crosses the isoquant through a non-optimal input bundle.   If at that crossing point the isoquant is steeper, which direction along the isoquant leads to lower cost.  Your goal here is to tie the graphical approach to the numerical approach.  They are different representations of the same idea.

Only after doing the above should you plot the isocost through the cost minimizing bundle.  The conditions that characterize the optimum are best understood as being when the conditions for a non-optimal bundle don't hold.

Perfect Complements

Joseph wrote:

"LUCAS has fixed money income, I which spent two goods X and Y. The prices of X and Y are fixed. Lucas,s Utility is based on following utility function. U(x,y)= min(4X,16Y). His income share for X is SX where Sx = PxX/I
and his income share for Y is Sy, where Sy = PyY/I
a: derive his demand function for X and Y.
b; using your answers from a, derive the own-price elasticity of demand, cross-price elasticity of demand and the income elasticity of demand for X and Y.
Thanks your help is appreciated."

My response:

This looks like a problem from a textbook.  My preference is to not provide answers to those but only some general guidelines to help you think it through.  Here I will content myself with part a of the question question.  Part b asks you to do some grinding based your answer to part a.

The question is asking about choice for a particular class of preferences called "perfect complements" or fixed proportion preferences or Leontief preferences, after the economist Wassily Leontief.  It turns out that the demands generated by these preferences have no substitution effect.  The have only an income effect.

This first graph gives an idea of what the indifference curves look like when the proportions are 1:1.

The indifference curve has a right angle at the 45 degree line.  Above and to the left of the 45 degree line Good 2 is redundant and Good 1 is scarce.  Then utility is determined by the amount of Good 1.  Below and to the right of the line Good 1 is redundant and Good 2 is scarce.  Then utility is determined by the amount of Good 2.   In the problem posed the proportions are not.  1:1.  It looks like they are either 4:1 or 1:4.  Figuring out which is something you'll need to determine.

The second graph should give you and idea about how to solve for the demands.  

Since it is always optimal to consume the goods such that neither is redundant, the choice will always end up on the dashed line, ergo the fixed proportions.  The choice will also be on the budget line.  That gives two linear equations that must be solved to get the demands.

The effect of a price of X change

Elly asked:

Hi Prof, can i ask you some questions regarding the budget line and indifference curve? When price of X falls, and X is normal, does it mean that normal good will always be on the right side of the budget line that has been separated by a point C? If that is the case, when price of X rises, and X is inferior, does the inferior good always falls on the right of budget line? Because from what i've known, when price of X rises, the budget line will rotate to the left from the original budget line, which means income decreases, so people will buy more inferior good, so inferior good will be on the right. Is that always true? I have also seen a few cases where the inferior good is on the left side even though Price of X rises and i cannot understand. Thank you for taking time to read my enquiries.

My response:

First, let's stick to the case where the price of X rises.  Afterward, the case where the price of X
falls can be worked through by doing the same analysis but in reverse.  Next, note that there are two effects to consider from a price change - a substitution effect and an income effect.  Let's consider those effects separately and then put them together.

Substitution effect

An increase in the price of X causes an increase in the relative price of X, because the price of Y has remained constant.  When a good's relative price has risen the substitution effect says less of the good (move to the left in the way Elly expresses it above).

Income effect

An increase in the price of X rotates the budget line inward around the Y intercept.  As long as some X was being consumed before the price change, that bundle is no longer affordable so this change means a reduction of real income.  The consequence of that income change on the amount of X consumed depends on whether X is normal or inferior. When X is normal the reduction of income leads to reduced consumption of X (again, that is a move to the left).  When X is inferior, the reduction of income leads to an increase in the consumption of X.

Overall

The substitution and income effects support each other when X is normal.  In this case the overall is to have less X consumed.  When X is inferior, however, the income effect offsets the substitution effect.  As an empirical matter we think that mainly the overall is determined by the substitution effect, so there still will be less X.  But it is logically possible for the income effect to win out, in which case the good is called a Giffen Good, named after the Scottish economist Sir Robert Giffen.

Question from Tracey

Tracey asked:

Hi Professor Arvan,
I have viewed some of your video's on YouTube, and was wondering if you provide people with the Excel spreedsheet's? I am particularly interested in having a closer look at the one you used for the Expected Utility Hypothesis video.

All the spreadsheets can be found here.  Each of the workbooks there must be downloaded and used in Excel.  There are multiple spreadsheets per workbook.  The name of the spreadsheet should coincide with the name of the video.

https://docs.google.com/#folders/0Bz9kxuxY68EJOTczMWM3ZmMtNmExYS00NDFjLTg4NTQtYTRlMjE3MWE5MjIx

Question from Mark

Mark asked:

First off, thanks for your videos. They help.My question:Assuming I have 0 means lottery X, (-6 with 1/2 probability and 6 with 1/2 probability) and utility function u(w)=w for w<=10 and u(w)=1/2w+5 for w>=10
Can I apply the Arrow-Pratt approximation of pi(w;X)=1/2 (sigma)^2A(w)?
My hunch is no since A(w)=0 in either case of u(w)...i think?? My question is what does A(w)=0 mean? and why does Arrow-Pratt not work here?

My response - The Arrow-Pratt Measure applies to utility functions that are twice differentiable.  In the example above the utility function is piece-wise linear, with a kink at 10.   It is not differentiable at the kink, so it is outside the class of functions for which the measure is intended.  Alternatively, if you prefer, when w is not 10, the individual is risk neutral (for small gambles).  When w is 10, the individual is infinitely risk averse.

How this should work

Students post their questions via the Web form on the third tab.  They also provide links to any ancillary information they believe is relevant.  The Prof takes that information and uses it as the start of a blog post.  Then The Prof provides a response that might be plain text, perhaps include a diagram or other support visuals, and might also be a short video.

All students, the student posing the question initially and any other student as well, can ask follow up questions using the comments area in the blog post.