### Tuesday, April 21, 2009

## PS 1 Hint

I've received the same question from several people, so let me offer a brief refresher on solving for optimal quantities given an MRS and a budget constraint.

Typically, these types of problems ask you to solve for the optimal amounts of two variables. In the labor supply world, we are interested in total consumption (C) and total Leisure (L).

Recall from algebra that to solve for two unknowns, we need two equations. We have two. We know that MRS = w and we know that the agent faces a budget constraint (wT + a = C + wL -- this says the total value of time and unearned income = the total value of consumption and leisure). With these two equations we can solve for the two unknown quantities.

Assuming an MRS of C/L and the above budget constraint, we can solve for C in terms of L (and the other variables):

C = wL (from the first equation)

C = wT + a - wL (from the budget constraint)

Since the left hand side of both equations is C, we know that the right hand sides are equal:

wL = wT + a - wL

or (solving for L)

L = (wT + a) / 2w

Which in turn (plugging back into the equations above) implies

C = w * [(wT + a) / 2w]

And the total number of hours worked = T - L

One other thing. On the problem set, I note that I don't penalize late assignments until they impose costs on me. Late problem sets typically impose two potential costs. First, it is seriously annoying to have to grade assignments after having graded all the others because the grade rhythm is gone. Second, for similar reasons, it is annoying to answer student questions about the problem set well after having dealt with others students' questions.

Thus, if you want to avoid penalties, you should (a) turn in your assignment before I start grading and (b) not ask for any problem set related help after the deadline. Technically, I can start grading anytime after the assignment is due, but grading is not one of my favorite activities so I am seldom eager to do it.

Typically, these types of problems ask you to solve for the optimal amounts of two variables. In the labor supply world, we are interested in total consumption (C) and total Leisure (L).

Recall from algebra that to solve for two unknowns, we need two equations. We have two. We know that MRS = w and we know that the agent faces a budget constraint (wT + a = C + wL -- this says the total value of time and unearned income = the total value of consumption and leisure). With these two equations we can solve for the two unknown quantities.

Assuming an MRS of C/L and the above budget constraint, we can solve for C in terms of L (and the other variables):

C = wL (from the first equation)

C = wT + a - wL (from the budget constraint)

Since the left hand side of both equations is C, we know that the right hand sides are equal:

wL = wT + a - wL

or (solving for L)

L = (wT + a) / 2w

Which in turn (plugging back into the equations above) implies

C = w * [(wT + a) / 2w]

And the total number of hours worked = T - L

One other thing. On the problem set, I note that I don't penalize late assignments until they impose costs on me. Late problem sets typically impose two potential costs. First, it is seriously annoying to have to grade assignments after having graded all the others because the grade rhythm is gone. Second, for similar reasons, it is annoying to answer student questions about the problem set well after having dealt with others students' questions.

Thus, if you want to avoid penalties, you should (a) turn in your assignment before I start grading and (b) not ask for any problem set related help after the deadline. Technically, I can start grading anytime after the assignment is due, but grading is not one of my favorite activities so I am seldom eager to do it.

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