General Equilibrium - Illustration (Lexicographic Preferences)

By Aleesha Mary Joseph

Consider a 2*2 pure exchange economy (ie 2 goods & 2 consumers).

The endowments are –  Individual 1 (8,8)
                                   Individual 2 (2,2).

Utility functions are as follows:

(1) Individual 1 has lexicographic preferences with respect to X. That is
  The bundle (x1 , y1) will be preferred to (x2 , y2) when x1>x2.
   But if x1=x2, then the bundle (x1 , y1) will be preferred to (x2 , y2) if and only if y1>y2.

(2) Individual 2’s utility function is given by U(x,y) = x^2 + y^2

Now our task is to find the contract curve and the competitive equilibrium.

SOLUTION

(I) CONTRACT CURVE

First convince yourself that we cannot construct indifference curve (IC) for individual 1’s preferences. Because there are no 2 distinct bundles between which individual 1 will be indifferent.

Now let’s figure out how 2nd individual’s IC looks like.

For example let’s plot IC that represents 4 utils. Ie x^2 + y^2 = 4. Note that this is actually the equation of a circle centered at (0,0), with radius 2. Also since we are bothered about only positive values of x & y, IC representing 4 utils will be represented as follows:

Now let us put everything in our edgeworth box as in the following figure(please google or refer any standard microeconomic textbook if you don’t know what an edgeworth box is). The dark point represents the endowment.

Now let us find the pareto efficient points. Pick any IC of individual 2 (say the red one). Now consider the point A. If we move along the IC in the direction given by the arrows, then individual 1 is getting better off and individual 2 is not getting worse. Hence the point A is not pareto efficient. Using the same logic we can argue out that there is only 1 pareto efficient point along the red IC and that is B. Now convince yourself using the same argument that all the points along O1B and BO2 are pareto efficient. Hence the contract curve is O1B and BO2.

(II) COMPETITIVE EQUILIBRIUM

Notations : X11 – quantity of good 1 demanded by individual 1

      X12 – quantity of good 2 demanded by individual 1

      X21 – quantity of good 1 demanded by individual 2

      X22 – quantity of good 2 demanded by individual 2

       Mi   - Income of individual i.

First of all let us write down the demand function for individual 1

(X11, X12)  =  (M1/P1 ,  0)  { Because for individual 1 only consumption of good 1 matters. }

The demand function for individual 2 is derived using the same logic as mentioned in the previous file (similar to the max function in the file).

        (X21, X22)  =  (M2/P1 ,  0)                                    when P1 / P2 < 1

      (M2/P1 ,  0)  or (0, M2/P2)                                 when P1 / P2 = 1

        (0, M2/P2)                                                          when P1 / P2 > 1

Normalize P2 = 1.

Now,

Since individual 1’s endowment is (8,8), income of individual 1 M1 = 8P1 + 8.

Since individual 2’s  endowment is (2,2), income of individual 1 is M2 = 2P1 + 2.

Note that since individual 1 consumes only good 1, all of good 2 has to be consumed by individual 2.

Hence from individual 2’s demand given above, it is clear that P1 / P2 has to be greater than or equal to 1. Since P2 is normalized, P1 ≥ 1.

Note that the budget line should pass through the endowment. 

CASE I

Suppose the budget line is the green line given below.

On extending the green line we get the following figure:

Convince yourself that:
• The extended green line is the budget line faced by individual 1.
• Given this budget line, individual 1 will consume the bundle P.
• This implies that in the economy there will be excess demand for good 1 as individual 1’s consumption is outside the edgeworth box.

CASE II

Suppose the budget line is the RED line given below.

On extending the red line we get the following figure:

Convince yourself that:

• The extended RED line is the budget line faced by individual 2.

• Given this budget line, individual 2 will consume the bundle Q.

• This implies that in the economy there will be excess demand for good 2 as individual 2’s consumption is outside the edgeworth box.

CASE III

Suppose the budget line is the BLUE line given below.

Convince yourself that when the budget line is the blue line, it is optimal for both individuals 1 and 2 to consume the bundle B. Hence the price ratio associated  with the blue line will be the competitive equilibrium price ratio. Since (8,8) is the coordinate of the black dot and (10,0) is the coordinate of B, the euilibrium price ratio P₁/P₂ = 4.

From the above three cases note that competitive equilibrium can occur only in the third case and in the other 2 cases markets do not clear.

On summarising, competitive equilibrium allocation is (10,0) for individual 1 and (0,10) for individual 2 with competitive equilibrium price ratio being  4.

(Aleesha Mary Joseph graduated from St. Stephen's College in 2013. She is currently pursuing MA in Economics at Delhi School of Economics)

General Equilibrium - Illustration (Max and Min Functions)

By Aleesha Mary Joseph

FINDING OPTIMAL DEMAND FOR GOOD X AND GOOD Y WHEN THE UTILITY FUNCTION OF THE CONSUMER IS
                                                
  U = min{ X+2Y , 2X +Y }  

Subject to the budget constraint  Px X  +  Py Y  =  M  
where Px is the price of good X and Py the price of good Y.

Step 1

Let us first figure out the indifference curve(IC) pertaining to this particular utility function. We know that IC is the combination of all those bundles (x,y) which gives same utility to the consumer. So let us keep utility constant at say 10. Then now our task is to plot  min{ X+2Y , 2X +Y }  = 10 on the X-Y space. If min{ X+2Y , 2X +Y }  = 10, then either (i) X+2Y  = 10  OR  (ii) 2X+Y  = 10.  Let us plot (i) & (ii) on the same graph:

Note that (i) & (ii) intersects when X=Y. And slope of (i) is ½ and slope of (ii) is 2.

 Now note that the yellow region corresponds to X + 2Y < 10. Also at any point on the red line 2X + Y =10. These statements imply that at any point on the red line min{ X+2Y , 2X +Y }  = min{ <10 , 10 }. ie min{ X+2Y , 2X +Y } < 10. This implies that the red line is not part of the IC representing 10 utils. 

 Similarly the orange region corresponds to 2X + Y < 10. Also at any point on the blue line X + 2Y =10. These statements imply that at any point on the red line min{ X+2Y , 2X +Y }  = min{ 10 , <10 }. ie min{ X+2Y , 2X +Y } < 10. This implies that the blue line is not part of the IC representing 10 utils. 

Note that the grey region corresponds to X + 2Y > 10. Also at any point on the purple line 2X + Y =10. These statements imply that at any point on the purple line min{ X+2Y , 2X +Y }  = min{ >10 , 10 }. ie min{ X+2Y , 2X +Y } = 10. This implies that the purple line is part of the IC representing 10 utils. 

Similarly, the blue region corresponds to 2X + 2Y > 10. Also at any point on the orange line X + 2Y =10. These statements imply that at any point on the orange line min{ X+2Y , 2X +Y }  = min{ 10 , >10 }. ie min{ X+2Y , 2X +Y } = 10. This implies that the orange line is part of the IC representing 10 utils

Thus from the above arguments it is clear that the points on the red and blue lines are not part of the IC that gives 10 utils. That is those points do not solve min{ X+2Y , 2X +Y }  = 10. Therefore let us erase the red and blue line segments from our figure. While the points on the purple and orange lines solve min{ X+2Y , 2X +Y }  = 10.

 The figure ABC represents all those bundles (x,y) which satisfy min{ X+2Y , 2X +Y }  = 10. Hence ABC is the IC which represents 10 utils. Note that slope of BC is ½ and slope of AB is 2.
The Indifference map is as follows:

Now let us consider various price ratios and figure out where the consumer will consume.


Step 2

(1) If Px/Py < ½ then budget line would look like the red line in the following figure:

The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at E. At E consumer spends his entire income on good X and consumes zero of the other good.

(2) If Px/Py >2 then budget line would look like the red line in the following figure. The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at F. At F consumer spends his entire income on good Y and consumes zero of good X

                                     

(3) If ½<Px/Py <2 then budget line would look like the red line in the Figure 10 The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at G. We know that at the kink G X=Y. Substitute this in the budget constraint. Then we will get the optimal consumption as X = Y = M/(Px + Py).

                                      

(4) If Px/Py = ½, then any point on BC is optimal. And any point on BC will satisfy X + 2Y = 10 & the inequality X≥Y.

                       

(5) Similarly when Px/Py = 2 any point on AB is optimal. And any point on AB will satisfy 2X +Y = 10 & the inequality Y≥X.


To summarize we can write demand for x & y as follows:
(x,y)  =   (M/Px, 0)                                                                          when Px/Py  < ½
              All (x,y) such that X + 2Y = 10 & X≥Y                                  when Px/Py  =½
             (M/(Px+Py), M/(Px+Py))                                                    when ½<Px/Py<2
              All (x,y) such that 2X + Y = 10 & Y≥X                                  when Px/Py  =2
              (0 , M/Py)                                                                        when Px/Py  >2



FINDING OPTIMAL DEMAND FOR GOOD X AND GOOD Y WHEN THE UTILITY FUNCTION OF THE CONSUMER IS
                                                  U = max{ X+2Y , 2X +Y }  


Using the same technique of analysis as in the previous question, the IC map of the max function would be as follows:

Exercise: Consider ABC is the IC representing 10 utils. Then Find the coordinates of the points A & C. Now verify that the slope of AC is 1. Convince yourself that for any given IC the slope of the line segment joining the end points of the IC (like A & C) is 1.

1) When Px/Py<1ie slope of the budget line (red line) is less than 1.

Clearly consumer will optimally choose the bundle C.



2) When Px/Py>1ie slope of the budget line (red line) is greater than 1.

In this case optimal consumption will be at A.


3) When Px/Py=1ie slope of the budget line (red line) is equal to 1.

In this case optimal consumption will be at A & C.


To summarize we can write demand for x & y as follows:
(x,y)  =   (M/Px, 0)                                                                        when Px/Py  < 1
               (M/Px, 0) or (0, M/ Py )                                                  when Px/Py  =1
               (0, M/ Py )                                                                     when Px/Py  >1


(Aleesha Mary Joseph graduated from St. Stephen's College in 2013. She is currently pursuing MA in Economics at Delhi School of Economics)