Tuesday, March 11, 2014

Estimating Demand Elasticities and Compensating Variation

A student posed the following question:

Hi Respected Sir,
I am writing a report on Compensating Variation (CV) in case of more than two goods say 8 goods. My question is how can we estimate it in case of more than 2 goods.
Another Question is How can we write the equations for 8 commodities in Almost Ideal Demand System (AIDS) to calculate Own , Cross and Expenditure Elasticities of demand.
In which software both of the methods can be calculated?
I am waiting for your reply.
Thanks in anticipation

My response:

This question is more econometrics than it is intermediate micro.  So I am only going to provide a partial response and stick to the economic theory part, though I must say I'd only teach what I discuss below at the graduate level. 
The traditional approach to consumer choice is to start with preferences, specified by a utility function, u, and combine that with a budget constraint, that depends on a price system p = (p1,p2,...,pn) that specifies the prices of each good or service, and the consumers income y.  Together the price system and income determine the budget set.  The consumer's choice, or demand, or optimum, call it x*(u,p,y) solves the problem of maximizing utility subject to the budget constraint.   
There is an alternative approach called the duality approach, which is useful conceptually and in laying the foundation for the econometrics.  Two value functions are determined.  One, measured in dollar terms, is called the Expenditure Function.  It is analogous to the Cost Function developed in the theory of the firm.  The expenditure function maps indifference curves (or when there are more than two goods, level sets of the utility function) and price systems into an expenditure level - the least expenditure it takes to reach that indifference curve at the given prices.   The other value function, measured in utility terms, is called the Indirect Utility Function.  It maps budget sets into utility levels.  Alternatively, it gives the utility attained at the consumer's choice. 
One of the powerful results from duality theory is that you can recover the consumer demand's from these value functions.  The compensated demands (these measure the substitution effect only) are given by the first partial derivatives of the Expenditure Function with respect to the specific price.  The ordinary demands can be obtained in a similar way from the Indirect Utility Function, though the result, known as Roy's Identity, is a bit more complicated.   
Let me close with the little I know about the Almost Ideal Demand System.  Deaton and Mulbauer start with the Expenditure Function, express it in log form, and then linearize it locally, assuming it is some average of the expenditure at subsistence (the worst possible point) and bliss (the best possible point).  This makes it suitable for estimation. 
Good luck on your paper. 

Thursday, September 5, 2013

The effect of a tax - curve shifting

Vu asked:

Dear professor Arvani have a questionWhy when a unit tax is levied directly on consumers, this make the demand curve shifts downwards? and what could be the causes of this shift ?
Thank you very much

My response:

The tax is levied in some market for a good or service.  Typically, buyers pay some of it and sellers pay some of it as well.  The tax creates a wedge between what the buyer pays and what the seller receives, with the difference being the tax.  If in the diagram the price represented is the seller's price (so the supply curve remains the same) then the demand curve shifts down by the amount of the tax.  In this way if you add the tax to the seller's price, you get the price really does pay so the quantity demanded should be exactly what it was before at this higher price.

Monday, July 8, 2013

Comparative Statics of Consumer Choice

Judy asked:

using well labele diagrams show the total effect,income effect and substitution effect due to a price fall of good x1 assuming that the consumer baskets has only good x1 and x2. The price of good x2 remains constant.

My Response:

I encourage you to watch my video on income and substitution effects.  There the goods are labeled x and y (instead of x1 and x2) and it is for a price increase rather than a price decrease.  I suspect you covered something similar in your class and now your instructor wants you to think through what happens when the price change is in the opposite direction.

From my video in the description there is a link to the spreadsheet.  Go to the tab labeled Income and Substitution Effects. There is a button called Raise Price of Good X.  Push on that and look at the cell next to the button.  It has positive values.  If you type into that cell a negative value, say -15, you will get the graph for a price reduction of Good X.  Note that because this was designed for a price increase, the new budget line doesn't extent all the way to X axis.  It should.

Also, your instructor may feel I'm doing your homework for you and not be happy about it.  So if you pose another question of this sort to Ask The Prof, please also include what you've tried to do in response.  This way I can help you without giving away the answer to the question.

Tuesday, April 30, 2013

Produce or Not in the Short Run?

Joe R. asks:

Working on  a question with a full chart to reference but am lost where to get the date for the answer. The question ask about the product price of $56, will this firm produce in the short run.?

My response:

There is a fairly complete analysis of the general issues in the video Short Run Cost.  The question is asking, in effect, whether there is any output level were Average Variable Cost is below $56.  If the answer is yes, then producing at that output level nets some producer surplus (the difference between revenue and variable cost), so it makes sense to produce.  If not, then it is best not to produce.

Note that the fix cost is not relevant for this calculation.  It is sunk in the short run and therefore must be paid regardless of whether production occurs.  

Your chart that you refer to may not have Average Variable Cost broken out, in which case you must compute it by dividing Total Variable Cost by Output.  If it is broken out, then it is simply a matter of eyeballing it to see if it ever is below $56.

Thursday, April 18, 2013

On the Weak and General Axiom of Revealed Preference

John asked the following:


"I have two related questions on Choice. 
I know that we can satisfy WARP but have nevertheless a violation of GARP. My question is if we can have a situation where WARP is violated, but GARP is satisfied? 
Secondly, from the definition of GARP it is always spoken of a bundle being revealed preferred to another bundle through a chain or ""sequence"" of revealed preferences. My question is, if this defined ""sequence"" can consist of only two observations, so that we have actually a direct revealed preference after all?  In other words, does ""revealed preferred"" include the case of ""direct revealed preferred""?"

My response:

The quick answers are, to the first question, no, and to the second question, yes.  In other words, GARP implies WARP and the chain can have only two elements, which is WARP directly.

A longer response would include making a comparison between revealed preference and the usual assumptions made about preference.  These are about a preference relation, R.  xRy is then read as, x is preferred to or indifferent to y.   So from R one can also define P and I by:

  • xPy if xRy and not yRx.
  • xIy if xRy and yRx.


There are three "logical" assumption about preference orderings.

R is complete.  For every x and y in the Consumption set (the set of possible consumption bundles) either xRy or yRx.  This means comparisons can be made between any two consumption bundles.  Note that neither P nor I are complete.

R is reflexive.  For every x in the Consumption set xRx.

R is transitive.  For every x, y, and z in the Consumption set, if xRy and yRz then xRz.

These properties allow one to define a choice, provided the choice set is closed.  In this case if the choice set is C then x in C is a choice (a maximal element under R) if xRy for all y in C.  Note that there is no greatest number less than 100.  If you posit it is 99, then 99.9 is greater and you can always add an additional 9 to the right.  So for a choice to exist, the choice set must be closed.  100 is the greatest number less than or equal to 100.  The choice set being closed means it includes its boundary.

To these logical assumptions, one usually adds an economic assumption - monotonicity or more is preferred to less.  This assumption rules about satiation points as well as "thick indifference curves,"  The upshot of this assumption is that when the choice problem is given by a budget set, the choice will always be on the budget line, never inside the line.

A further assumption that is frequently made is that preferences are Convex, which gives indifference curves their usual shape.  This is done do when the Budget environment changes in a small way, the choice also changes at most in a small way.  Or, if you prefer, the demands are continuous function of the budget environment.

Now, with all this machinery, what does WARP get you?  In this way of thinking, WARP is equivalent to completeness, reflexivity, and monotonicity.  You need GARP to bring in transitivity.  There is also something called SARP, the strong axiom.  It is WARP plus the assumption that preferences are strictly convex, so the choice is always unique.

I hope that helps.


Sunday, December 16, 2012

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.


Wednesday, December 12, 2012

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.

Wednesday, November 14, 2012

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.

Tuesday, November 13, 2012

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.


Saturday, October 20, 2012

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.