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Facing a little bit of a problem with this questions, did a similar one BUT the utility function was not linear and got MRS dependent on goods (was not just a number) - here I am at a loss.

The question: There are two individuals A and B, each having one unit of time which can be used either for leisure or labor (to produce a good to consume). Their utility functions are respectively:

uA(lA,cA) = 6(1-lA) + cA and uB(lB,cB) = 10(1-lB) + cB

Where 1-lA/B is the leisure time for A/B. Furthermore, we know that A can produce 4 units of the consumption good per 1 unit of labor time, and B can produce 24 units of the consumption good per 1 unit of labor time.

The last information I transformed into:

cA = f(lA) = 4lA and cB = g(lB) = 24lB

(where f and g are some production functions)

I am being asked to find the set of Pareto-optimal allocations of consumption and labor.

I assume I need to start with MRSA = -6 (or 6? never know when to use negative/positive?) and MRSB = -10. However, I am not sure what piece of information I can get from that? I assume that indifference curves are lines, right? And that they will always cross, which then would imply that the set of Pareto-optimal points is on the boundary of Edgeworth box?

On the other hand, these guys maximize their utilities so that:

max 6(1-lA) + cA = max 6(1-lA) + 4lA = max 6 - 2lA

and

max 10(1-lB) + cB = max 10(1-lB) + 24lB = max 10 + 14lB

meaning that lA = 0 and lB = 1. That is, A will spend his 1 unit of time for leisure: 1-lA = 1-0 = 1 and B will spend his 1 unit of time for labor.

Two other questions are:

  1. Find Pareto-optimal allocations that are also envy-free.
  2. How would the analysis go when A could produce 7 units of good per 1 unit of labor time.

But I think I should be able to answer them, after some hints for the first part of the question. ;)

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Hint 1: So I think you are making a mistake by substituting their production technology into their utility: when you went from $\max 6(1-l_A)+c_A$ to $\max 6(1-l_A)+4l_A$, by doing this, you are assuming that $A$ only consumes what she produces.

Hint 2: Let me give you a Pareto optimal allocation: $B$ spends his $1$ hour working to produce 24 units of consumption and gives them to $A$ who spends her 1 hour in leisure. This allocation is feasible (there are enough resources to do it, given the technology), and it is Pareto optimal because $A$ is the happiest, so there is no way to improve $A$, and in order to improve $B$, you would have to decrease $A$'s utility. Of course, this is allocation is not envy-free, but it is a Pareto optimal allocation.

Hint 3: I would first find all the possible allocations in order to then see which ones are Pareto, and then which ones are envy-free.

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I don't think the edgeworth box will help here since there is only 1 good in the economy that is exchangeable. According to 1's utility function, he would always want to allocate his entire endowment of time to leisure. Thus, any allocation in which he consumes positive amount of good while consuming less than 1 labor is not efficient, because he can always opt to increase his utility by converting that positive amount of good to leisure and increase his utility. Similarly, you can reason for B. The set of pareto efficient allocations is such that (1-La, Ca, La)

    • 1 consumes (1,C*,0),
      • 2 consumes (0,24-C*,1),
      • enter image description here
    • 1 consumes (0,0,1)
    • 2 consumes (0,28,1)

1 will not envy 2's allocation only if enter image description here and B will not envy A's allocation if enter image description here. Hence there are no envy free allocations. The analyses changes when A can produce 7 units from 1 unit of labor. Now, A will always prefer to spend all his time on labor. Thus any allocation in which A consumes positive amount of leisure is not efficient. The set of pareto efficient allocations now is such that

  • 1 consumes (0,C*, 1)
  • 2 consumes (0,31-C*,1)
  • enter image description here

A will not envy B's allocation only if enter image description here B will not envy A's allocation only if enter image description here Therefore , the only envy free allocation is (0,31/2,1),(0,31/2,1).

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