1 - Modeling an economy - Introduction

Question 1

What’s the set of consumers?

Answer 1

i = 1….I

Question 2

What’s the set of firms?

Answer 2

j = 1…..J

Question 3

What’s the set of goods?

Answer 3

l = 1…..L

Question 4

What’s the total initial endowment of good l ?

Answer 4

ωl >= 0

Question 5

Whats the consumption bundle of consumer i?

Answer 5

Xᵢ = (X1ᵢ, …., XLᵢ)

Question 6

Whats consumer i’s ownership of firm j?

Answer 6

θᵢⱼ Є [0, 1 ]

Question 7

When is good l an input or output for firm j?

Answer 7

Input: Ylⱼ < 0
Output: Ylⱼ > 0

Question 8

When is an economic allocation feasible?

Answer 8

If ΣXlᵢ <= ωl + ΣYlⱼ

Question 9

What is the definition of pareto optimal?

Answer 9

if there is no other feasible allocation (x’, y’) such that

uᵢ(xᵢ’) >= uᵢ(xᵢ) for all i & uᵢ(xᵢ’) > uᵢ(xᵢ) for some i

Question 10

What is the profit maximization condition for a competitive equilibrium?

Answer 10

yⱼ* = P* yⱼ (maximizing yⱼ)

Question 11

What is the utility maximization condition for a competitive equilibrium?

Answer 11

xᵢ* = uᵢ(xᵢ) (maximizing x)

subject to Pxᵢ = Pωcᵢ + Σθᵢⱼpyⱼ

Question 12

What is the market clearing condition for a competitive equilibrium?

Answer 12

Σ Xlᵢ* = ωl + Σ Ylⱼ*

Question 13

You are given utility, endowment ω, ownership share θ and production function. How can you find out whether any given allocation is feasible? (5 steps)

Answer 13

1. You use the economic feasibility constraint
2. You set up constraint equations for each of the goods
3. You use the production function as constraint for how much you can produce given a certain input
4. You plug in the given allocations into the constraints
5. ALL constraints need to be fulfilled in order for the allocation to be feasible!

Question 14

You are given utility, endowment ω, ownership share θ and production function. How can you find out which allocation is pareto optimal? (6 steps)

Answer 14

1. You use the consumer’s utilty function
2. You set up constraint equations for each of the goods
3. You use the production function as constraint for how much you can produce given a certain input
4. You maximize the utility function (plugging in the constraints to have as few variables as possible)
5. You take the first order condition (first derivative equal to zero) and solve it for one variable
6. Plug in constraints to get other variables

Question 15

You are given utility, endowment ω, ownership share θ and production function. How can you find out whether any given allocation is pareto optimal? (3 requirements)

Answer 15

You check whether for any given allocation:

1. all input is used to produce the maximum output
2. all output is allocated with the consumers
3. the utility for the consumers is not 0 (i.e. with utility function u(x)=x1*x2, any allocation where x1 and x2 > 0 is better than where x1 or x2 are 0.)

Question 16

Explain the economic meaning of the assumption that utility functions ui are strictly
increasing.

Answer 16

This means that consumers have non-satiated preferences, i.e. that they always prefer to have more over less of a certain good c.p.

Question 17

Show that any allocation ΣXli < ωl + ΣYlj is not pareto optimal.

Answer 17

It is not pareto optimal because there is still goods (ωl + ΣYlⱼ - ΣXlᵢ) left that can be distributed to make some consumers i better off while keeping all other consumers the same.

Question 18

Consider a feasible production vector yⱼ Є Yⱼ. Under what condition lies this production vector on firm j’s production frontier?

Answer 18

The production vector is efficient if production cannot be increased without increasing the use of inputs.

Question 19

Argue that in this economy any feasible economic allocation for which there is some production vector yⱼ that does not lie on the firm j’s production frontier is not Pareto optimal.

Answer 19

Given y is not an element of the production frontier, it is possible to produce more goods with the same amount of inputs. That means it is possible to increase some consumer i’s utility without decreasing anybode else’s utility.

Question 20

How can there be several pareto optimal allocations with one consumer and multiple firms?

Answer 20

a linear indifference curve, linear and identical production functions