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)