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:
- Find Pareto-optimal allocations that are also envy-free.
- 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. ;)