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


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. ;)


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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