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)

My Essay for Rhodes

© Jalnidh Kaur

Dear applicant for the Rhodes Scholarship
Everybody is unique. You would know by now that a Rhodes essay is a highly personal one, and so essays are widely different across different individuals. The essay is nothing superficial, but a true reflection of one's real self - your dreams, aspirations and plans for the future. 

I have been receiving loads of requests to share my essay, I have already shared it with a few aspirants of the current year. Deep inside, I have been introspecting whether I have been promoting cronyism by this kind of selective sharing with people who are requesting me for the same. In the interest of equal opportunity and as a hardcore proponent of knowledge sharing, I am sharing it publicly. Please find the essay below. 

Writing the essay last year was a truly wondrous experience. I thought it involved long travails into my self and I became much clearer and maturer after having written it. Often the only person people don't know much about, is their own selves. And the essay asks you do just that hard chore - knowing yourself and expressing yourself. I hope you too will succeed in the task and will enjoy the journey.

Good wishes with your application.
Regards
Jalnidh Kaur

It has been a strange summer. While the temperature shot up like never before, I was juggling between two diametrically opposite worlds. One, a luxuriously conditioned and hi tech workspace where things moved fast and the surroundings were spic and span, the other was a mess - dust, heat, garbage and stink; streams of sweat would rush down my spine while the sun would scorch my face.

The first was that of a regional stock exchange where the AC chilled me to the bone while the second was the very visible underbelly of my city. The first was more an act of courtesy to a boss; the second was an act of personal choice.

Paradoxes abound in our little world. As an undergrad student, I have always been awed at the contradiction between a conditioned and an unconditioned weather, between the classy basketball court at my school and the rickety one at a government school, between public sector and private sector, between affluence and penury, between North Delhi and South Delhi.  Of course perfect equality and justice are just ideal goals and chimeras, even Rawls would acknowledge the impracticality of the same. What philosophers, economists and policy makers strive to achieve is a less unjust society, all policies are attempts to remove clear injustices to an extent possible. 

Fascinated by one of the attempts to eliminate injustices in the domain of health, I chose to independently study the RasthriyaSwasthyaBimaYojana – RSBY (National Health Insurance Scheme) in my summer this year. The scheme provides a handsome health insurance cover to the population below poverty line to deliver social security. I interviewed these beneficiaries to study how their lives changed.

While, philosophical parleys with peers at college always tend to bend to one side, either extreme libertarian or extreme socialist; I have learnt through this study how private sector, government and the poor can all come together in a symbiotic manner and create mutual benefits.  Research has been a delightful discovery and the search for truth has instilled a new knowledge hunger in me – to find out bits of truth from the ground and supplement the mathematical rigor of economics with empirical evidence. Most importantly, I have learnt how technology and efficient delivery can create a workable incentive matrix and create social change.

Five years ago, I chose to opt for economics out of sheer enthusiasm of exploring it. I wanted to study a discipline which could understand the social complex of interrelationships objectively and evaluate it quantitatively. Over time, I have discovered the depths of the discipline and the implements it equips its follower with. Economics principles are like a toolchest. Through analytical frameworks and models, I feel equipped with more gear at the end of every lecture. The freedom it bestows on its student is just so enthralling – one could study anything from markets in education to health to nuclear physics. This limitless horizon has allowed me to explore my minor fields in depth along with my major. My philosophical inclinations along with a brush with Boulding’s views on going beyond the rudimentary and strait-jacket assumptions of the discipline have motivated me to integrate my everyday ruminations with the principles of economics.

Another interest which has over time closely integrated with my discipline is ‘education’. New constructions are in full swing in my locality. In the vacations following my board exams I started teaching kids of the construction workers who would romp aimlessly in streets every day. The bench under a shady tree in the park became our classroom. They loved shouting the English alphabet aloud as if trumpeting their first baby step towards literacy in the neighborhood. I told them what an 'echo' is – when you shout aloud, your voice hits the houses nearby and it returns. They shouted louder to hear their voices echoed by the newly built posh houses around, content to discover the power of their own vocal cords.

Over time this little initiative which we christened as ‘Éclair’ (Enlightenment), has blossomed into a country wide street school chain functioning in three states including Delhi. My school and college batch mates pitched in this voluntary initiative as guest faculty. Till date, we have been able to get ten kids 'enrolled' in regular schools– it is an achievement that rejoices me much more than any of my personal educational accomplishments. I have discovered my pedagogic inclinations through this initiative.

As I look ahead at the expanding horizon that economics has thrown open for me, I see myself doing an M. Phil in Economics from Oxford. This would equip me with more tools in my toolbox and allow me to understand the world of inequalities and injustices better. I have had a vibrant exposure to financial markets and live trading in a stock exchange during my internship and I have seen the world of poignant injustices. Capital markets offer huge dividends and bonuses, of a scale and size that people down the poverty lane, below the poverty line cannot even dream of. I aim to use my learning from the study of social sector schemes in developing countries and that about capital markets to arrive at logical solutions. An M.Phil in Economics will provide me an academic base and better foundational framework within which these inequalities could be examined. I shall be in a better position to assess various developmental strategies and propose better solutions, most importantly. A dedicated focus on academic research at Oxford will help me explore the world further and allow me to go beyond limited model-specific assumptions of the discipline.

Five years ago my essay on International Peace won me laurels at the University of Hawai’i and East West Center, USA. The essay ended with a note of conviction – Make history by design and not by accident. Today as I mull over these lines, I am of a firm conviction that disparities in our world could be bridged with value-based interventions and firm foundations in social science research.