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.