0
$\begingroup$

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. ;)

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.