IV. Summary of the constrained optimization problem facing households and firms – We will return to this later in the course.

A. Households: maximize their utility subject to their income constraints

  1. max U(X1,…,XN) s.t. = Income
    1. This problem produces individual demand functions for the goods X1,…,XN.
    2. Generalized to include time – savings functions
    3. Generalized to include leisure – factor supply functions (labor supply)
    4. Generalized to include risk

B. Firms: maximize their profits in two stages: minimize costs and then maximize profits

  1. min C(L, K) = w·L + r·K s.t. = f(L,K), for some.
    1. The solution to this problem gives the cost curves that you have studied in ECON 201 and 303; C(Q).
    2. This solution also gives factor demand functions (e.g. demand for labor).
  2. Given (1) then max P·Q – C(Q).

 

V. Exchange Efficiency – Pindyck and Rubinfeld chapter 16 is a good introduction to the Edgeworth Box and all of the efficiency conditions. Also Just, et al. pp. 14-18.

A. Use the Pareto criterion to rank social outcomes

  1. Pareto criterion: If it is possible to make at least one person better off by moving from state A to state B without making anyone else worse off then society should rank state B higher than state A. We say that B is a Pareto improvement over A, or B is Pareto superior.
  2. When there does not exist any feasible Pareto improvement from some state C, then we say that C is a Pareto optimum – it is not possible to make someone better off without harming someone else.
  3. We can compare two inefficient states and determine whether or not one is a Pareto improvement.
  4. A state that is a Pareto improvement does not imply that it is Pareto optimal.
  5. Many states of the world are Pareto non-comparable – one person is better off and another is worse off. (Another criterion is needed in these cases (e.g. compensation test).)
  6. Pareto optimum, in its purest form, implies nothing about equity or fairness.

B. Edgeworth box – graphical construct that shows all of the possible allocations of two goods between two individuals. (Can also be used to understand the allocations of inputs used in the production of two different productions.)

1.      Suppose there are two individuals who have the following endowments: I: (10, 8) and II: (5, 12). (We ignore at this point the production of the goods.)

2.      Individual preference mappings.

3.      Suppose that at the endowment points MRSX, Y for individual I is -1; and for individual II the MRSX, Y= -3.

4.      Combining preference mappings to form the Edgeworth box.

5.      Finding the Pareto improvements relative to the endowment point.

a.       Using each individual’s MRS

b.      Let II pay exactly his MRS; move along his indifference curve; I is better off.

c.       Let I pay exactly her MRS; move along her indifference curve; II is better off.

d.      Individual preference mappings

e.       Let the price, P, satisfy 1 < P < 3 (e.g. P =2). Both individuals are better off.

f.        All allocations within the lens-shaped region from the endowment point (including the boundaries) are Pareto improvements.

6.      Pareto inferior points relative to the endowment point are allocations below each of the individuals’ indifference curves that pass through the endowment point. Both are worse off relative to the endowment point.

7.      All other allocations (other than Pareto superior points and Pareto inferior points) are Pareto non-comparable (a limitation of the Pareto criterion).

8.      Pareto efficient points – all allocations for which there are no further Pareto improvements.

a.       Contract curve

b.      Exchange efficiency requires: MRSiX, Y = MRSjX, Y for all i ≠ j.