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.