Chapter 3: Allocation and Distribution

 

 

Exercise 3.1.

 

 

The Pareto optimal choice of Y by the allocation branch is given by the following problem:

 

If we solve the following system:

 

Then we will get the Pareto optimal amount of Y by the Allocation Branch:

 

 

 

 

Exercise 3.2

 

Since any monotonic transformation of the function  represents the same preferences, we know that the monotonic transformation does not change the marginal rate of substitution. On the other hand since all consumers have identical Cobb-Douglas utility functions, then there is a unique Y that satisfies the Samuelson conditions.

 

 

    (MRS for a Cobb Douglas utility function)

 

   (MRS for a utility function of the Bergstrom-Cornes form)

 

Suppose we have the following monotonic transformation:

 

 

. So if A (Y)= and Bi (Y)=0, then Vi is a utility function of the Bergstrom-Cornes form.

 

 

 

Exercise 3.3

 

 

 

A feasible allocation is given by .

 

Any allocation that maximizes the sum of utilities subject to the feasibility restriction is Pareto optimal.

 

The sum of utilities is:

 

 

 

 

From the solution of the problem  we get Y= and X=. The sum of utilities is then .

 

 

 

 

 So the UPS is the set

  .

 

We also know that what is making the distribution of utility among individuals is the distribution of the private good: . If  then  is Pareto Optimum. If , then  is not Pareto Optimum.

 

 

 

 

 

Exercise 3.4

 

 

 

a) The Samuelson condition is given by:

 

                                                                                                         (1)                                                          

 

The feasibility condition is:

 

                                                                                                                                (2)

 

 

If we plug (2) into (1) we get the unique amount of public goods that satisfies the Samuelson Condition:

 

 

 

b) The Pareto optimal points that do not satisfy the Samuelson condition are points that do not maximize the sum of utilities.

 

 

If it is the case that person  has no interest in person ’s consumption, then the problem that person  solves is the following:

 

 

From this problem we get:

 

 

.

When person  controls all the resources and maximizes his own utility, he still leaves something to person j by providing public goods, which means that the utility level of person j is not zero. Suppose the case where:

 

a) and

 

Then we have:

 

, , ,  and .

 

Any allocation  with is also Pareto optima but does not satisfy the Samuelson condition.

 

 

b) and

 

Then we have:

 

, , ,   and

 

Any allocation  with is also Pareto optima but does not satisfy the Samuelson condition.

 

Why? Because the only Pareto optimal that satisfies the Samuelson condition is .

Consider for example the allocation (,,).

Lets check if this allocation satisfies the Samuelson condition .

 

, which is false since by assumption .

 

c) The utility possibility frontier is formed by the following points:

 

 

(i) Set of points that comprises the utility distributions that result from allocations that are Pareto Optima and also maximize the sum of utilities:

 

 

 

 

Along the utility possibility frontier we know that:

 

 

 and .

 

 

 

So . This is a straight line with slope (-1) in the utility levels space.

 

 

(ii) Set of points that comprise the utility distributions that result from Pareto Optimal points that do not maximize the sum of utilities:

 

 

 

 

 

 

 

 

 

 

GRAFICO

 

 

 

 

 

The curve segment EF consists of the distributions of utility corresponding to allocations where . In this curve segment person 1 has all of the private goods and the amount of public good is less than the amount that person 1 would prefer to supply for himself.

 

Symmetrically, the curve segment BC corresponds to the utility distributions where person 2 detains all of the private goods and the amount of public goods is less than .

 

The points on the curve segments EF and BC belong to the utility possibility set but are not Pareto Optimal (if one follows the arrows one notices that it is possible to improve both agents’ utility).

 

The points on the interior of region (OABCDEF) can be achieved by means of an allocation in which .

 

 

d) We saw that if and  then the quantities of public goods were smaller than   those at Pareto Optimal outcomes:

 

 

 and .

 

If and then the quantities of public goods that do not satisfy the Samuelson condition but are Pareto Optimal are bigger than those that satisfy the Samuelson condition. Again the that maximizes the individual utility is not the Y that maximizes the sum of utilities:

 

 

 a.

 

 

Thus where  and , the  that maximizes Person 1’s utility is greater than the one that maximizes the sum of utilities.   

 

Notice that in this case, if  X1 +K 2 <0, then the marginal utility of person 2 for the public good is negative.   The Pareto optimal allocations that take into account person 2’s disutility for Y when she is poor, have less Y than the Pareto optimal allocation that ignores 2’s preferences and simply maximizes 1’s utility.

 

 

 

Exercise 3.5

 

The Samuelson condition is given by:

 

                                                                                                                                  (1)

 

The                                                                                         (2)

 

Plugging (2) into (1) one can get:

 

                                                                                                                    (3)

 

From the feasibility restriction we get that:

 

                                                                                                                              (4)

 

 

Substituting (4) into (3), we get the unique quantity of that satisfies the Samuelson condition as:

 

 

 

Exercise 3.6

 

a)      An allocation that maximizes the sum of individual utilities over all feasible allocations must be Pareto Optimal.

 

Suppose that the  allocation (X,Y) maximizes the sum of individual utilities over all feasible allocations. If this allocation is not Pareto optimal, there is another feasible allocation where each individual has at least as great a utility and someone has a higher utility.  But then the sum of the utilities in the second allocation would exceed the sum in the first allocation.  But this contradicts the assumption that (X,Y) maximizes the sum of individual utilities.

 

 

 

 

 

b) Where for all  any allocation that maximizes over all feasible allocations must be Pareto Optimal.  Proof:  Suppose not.  Then there is another feasible allocation where each individual has at least as high a utility and someone has a higher utility.  Any positively weighted sum of these utilities would be higher than   .  This contradicts the assumption that the original allocation maximizes .

 

 

 

 

 

 

Exercise 3.7

 

 Utility of each individual is given by Ui(Xi,Y)=Xi +Y1/2  .   The resource constraint is X1+X2+Y=3 with non-negativity constraints on all variables.

We seek to maximize 2U1+U2.   At the maximum, it must be that X2=0.  So we seek to maximize

2X1+3Y1/2  subject to X1+Y=3.  The solution is Y=9/16,  X1=3-9/16.

 

Exercise 3.8

 

a) For these utility functions be of the Bergstrom-Cornes form they must be represented by utility functions of the form:.

 

(i) Let   and

 

Then utility function belongs to the family of the Bergstrom-Cornes utility functions.

 

(ii) Let and

 

Then utility function is also of the Bergstrom-Cornes form.

 

b) Person 2 has positive utility for Y if  X2>Y and negative utility if X2<Y.  Person 2 has convex preferences.

 

 

 

c) To find the allocations that maximize the sum of utilities:

 

The problem to be solved is:

 

 

The solution is: and .

We see that there  is an interior solution only  if .  If this is not the case, then Y=W.

 

The sum of utilities is given by: