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)

General Equilibrium - Illustration (Cobb-Douglas)

By Aleesha Mary Joseph

Let's begin with an easy one to understand the concept of General Equilibrium.

Suppose, U₁(X₁₁ , X₁₂ ) = X₁₁ X₁₂ and U₂(X₂₁ , X₂₂ ) = X₂₁ X₂₂ .

Let (e₁₁ , e₁₂ ) = (10,0) & (e₂₁ , e₂₂ ) = (0,10).

Normalize P₂ to be equal to 1.

Then note that individual 1’s income M₁ = 10*P₁ + 0*1 = 10P₁.

Similarly individual 2’s income M₂ = 0*P₁ +1 0*1 = 10.

Since the utility functions are cobb douglas, demand functions can be derived by equating MRS with P₁/P₂ and substituting the relevant expression in the budget constraint.

The demand functions so derived will be as follows:

X₁₁ = M₁ /2P₁ = 10P₁/2P₁ = 5

X₁₂ = M₁ /2P₂ = 10P₁/2P₂ = 5P₁ (Because P₂ is normalized to 1)

X₂₁ = M₂ /2P₁ = 10/2P₁ = 5/P₁

X₂₂ = M₂ /2P₂ = 10/2P₂ = 5 (Because P₂ is normalized to 1)

The Market clearing condition says

X₁₁ + X₂₁ = e₁₁ + e₂₁

5 + 5/P₁ = 10. 

On solving this we get P₁ = 1. Hence the competitive equilibrium price ratio P₁/P₂ = 1

Convince yourself that the same price ratio will clear the market clearing equation for good 2 as well ie X₁₂+ X₂₂ = e₁₂ + e₂₂ .

Substituting P₁ =1 in the demand functions, we get the competitive equilibrium allocation

As (X₁₁ , X₁₂ ) = (5,5)

     (X₂₁ , X₂₂ ) = (5,5)

With competitive equilibrium price P₁/P₂ as 1.

The articles to follow will cover min/max utility functions and lexicographic preferences. Stay tuned.

(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

By Aleesha Mary Joseph

Hi friends,

This blog is already rich with advice for cracking DSE entrance. I would just like to emphasize on one thing i.e, doubt clearance. Because it is disheartening to see a question in the examination which you had marked as doubt in your preparatory material. It can even cost you a place in DSE.

No doubt is silly. If you find it hard to ask your teacher, then 

1)  Ask your friends or classmates. 

2)  Mail it to anyone who you think would know the answer. Doesn't matter even if the person is someone whom you have never met {may be you can message current DSE students (list available on DSE’s website) via Facebook}. 

3)  Post your questions on groups meant for economics entrance preparations. Even if you think your questions are very trivial, just post it, because there will definitely be many other students with the same doubt but are just shying away from posting the same. One such group can be found at http://economicsentrance.in/ (Group by Amit Goyal)

 Whatever the way is just make it a point to clear all your doubts before the D-day!!

For many students the general equilibrium(GE) section is scary and that is mostly because of their lack of knowledge on how to approach the GE problems. Trust me once you look at the technique GE becomes your cup of tea! 

I believe the following material on GE problems may be of some help to students who don’t attend coaching classes. I have skipped many steps. But feel free to ask for any further explanations. The material was made assuming that you already have some basic knowledge about GE set up. Please refer Nicholson & Snyder for conceptual clarity. The best way to master GE problems is by making your own problems with all sorts of utility functions and trying out in square and rectangular edgeworth box. If you go through the past year question papers you will get an idea of the various kinds of utility functions that the DSE faculty like. Most questions can be answered once you find the contract curve and competitive equilibrium allocation. So try to find the same for all sorts of weird looking combinations of utility functions and endowments: P Enjoy practicing and GE problems will be a cakewalk for you in the exam. Few questions can be found at http://econdse.org/pghosh-001/  :) 

General Set Up

Suppose individual 1’s utility function is represented by U1 (X11 , X12 ) and for individual 2 it is U2 (X21 , X22 ). 

Let the endowment for individual 1 be (e11 , e12) and for individual 2 it is (e21 , e22).

To find the competitive equilibrium allocation following steps are recommended:

1)  Find individual demands for good 1 and 2. ie X11 , X12  , X21 and  X22 . These quantities will be a functions of P1 / P2 . (or will be functions of P1 alone if we normalize P2 to be 1)

2)  Then equate X11 + X21 = e11 + e21 for this is the market clearing condition for good 1. i.e the total quantity of good 1 being demanded by individuals 1 & 2 should be equal to the total supply of good 1 available in the economy.

3)  On solving the above equation we will get the competitive equilibrium price ratio P1 / P2 .

4)  By Walras’s law the equilibrium price ratio obtained above will clear the market for good 2 as well.

General Equilibrium problems with step by step solutions will be covered in the articles to follow. Stay tuned.

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

Let's bell the CAT

By Suhani Popli

It was nearly a year and a half ago, when I was one of those many final year kids, juggling with the different ‘options’ that would help them build a future – that I decided to pursue the MBA degree. A dilemma that exists in most of our minds is that between a management degree from abroad versus that from within India. There is no end to this debate, but since I belong to the latter group, I would describe the process of getting through that in this piece of writing.

There are different exams held for the admission to a management institute in India, the popular ones being the CAT, the XAT, the NMAT, IIFT, exams specific to colleges, and so on. A gateway to the premier Business Schools of the country, the CAT, as an exam, does not really need an introduction, so I’ll skip that part. I’d rather use this space telling you what the different phases in the process of doing this exam are.

      1.  The Initial Phase

As one would expect, there are different ways to go about this. People can choose to do absolutely nothing, or study on their own selves, or join a coaching institute. If in the second category, I would suggest an aspirant to pick up study material from online/reliable offline sources. Trust me there is no dearth of either. However (and this opinion is strictly personal), I have always felt that the CAT requires not only a basic and complete knowledge of the relevant material for quantitative, logical and verbal ability, but also a grasp on the tips & tricks that could help accelerate one’s speed to solve.

My guides, both in terms of the preparation material and the teachers, came from the coaching institute I was enrolled in. And I genuinely believe that someone teaching me a particular concept would make it clearer to me, as compared to my reading it from a book. However, one is his/her own judge of what works best for the purpose of learning. Hence, the most important part in this phase is for an aspirant to pick the category that suits best.

Incase one does plan to join a coaching institute – honestly, there isn’t much of a difference between the names that exist in the market today. All of them have material and test series which are comprehensive. Thus, I would suggest to the aspirants to take advice from their seniors who must have attended such classes. Based on what they say (take views of students from at least two different places), make the final decision.

      2.  The Preparation Phase

Personally, my preparation was at the coaching institute I joined. However, for almost everyone I know, it is actually during the summer vacations when people put in their best effort to study. Personally, if I look back, I see that those two months were quite intense, and it was my coaching faculty who guided me through most of what I did. What is important in this phase is to identify the areas where one needs work and hence polish those well. There is no key to this, but to practise – regularly and efficiently.

As the end approaches, it becomes extremely significant to practise as many mock tests as possible. Also, practicing the CAT papers of the past five to ten years can be extremely useful. This serves a dual purpose – one of making the aspirants familiar to the exam pattern and the other of helping them learn how to manage time. Taking a step forward, it is more important for one to analyze mistakes post a mock test. Though this part is more time consuming than taking the exam itself, it ensures the avoidance of the same mistakes in future.

      3.  The D-Day

First, different myths float when one tries to choose a date – of the exam being easier on weekdays than on weekends, of taking the exam on a particular day, etc. – all of this, yes, ALL of it is absolute crap. Nothing such is true. The exam is such that there is a large common pool of questions designed each year, and questions are pulled out from this pool everyday in random combinations to form a question paper. Hence, one can never predict the difficulty level of any particular day or time slot.

Second, aspirants should make sure they look at all the 30 questions in each section. At various instances, the questions in the beginning tend to be difficult compared to the ones later in sequence. Thus getting stuck at initial questions can lead to very negative outcomes.

Third, keep calm and get done with the exam.

      4.  After the declaration of the results

Once the results are declared, successful aspirants get calls from the different business schools. The last step in the process entails preparation for the GD/WAT processes and the PI – i.e. the Group Discussion, the Written Ability Test and the Personal Interview rounds. The key to succeeding here would be to know two things well – one’s own resume and the current affairs. Infact, reading the newspaper is a habit that one should try and pick up early – since it also is helpful in the preparation of the verbal section of the CAT.

Like conventional CAT preparation pieces, I don’t want to end this article by stating a list of which business schools are the best in the country. One can easily Google and find that out. Rather, I would like to put forth the idea that it is important to think through the direction in which life is taking one. And whether it is a job, a master’s degree or an MBA – make an informed choice, one that you really want to do! All the best :)

Suhani Popli graduated from St. Stephen's College in 2013. She scored 98.2 percentile in CAT 2012 and is currently a student at IIM Kozhikode.