Need help solving a Lagrangian budget constraint problem. It’s due in two weeks for extra credit but the last exam average was a 42 and I need this in my life. So:
Static utility function of representative consumer:
MAX u(c, l) = ln (c) + B ln(l)
s.t. P*(1+tc)*c = W*(1-t)(24-l) - T
Where
B is a preference shock parameter
c is consumption
l is leisure
P is price
W is wage
T is a lump-sum tax
tc is a proportionate tax on consumption
t is the measure of proportionate labor income tax
Setting up the Lagrangian is simple as well as finding the marginal rate of substitution between consumption and leisure. My problem comes in finding the consumer’s optimal allocation. If there is some trick let me know. Will compare work if anyone is interested in helping.
For anyone who maybe doesn’t know economics but who grinds math basically I need to solve the system for the variables c and l.
Thanks brahs.
Are you literally just rearranging these to express in terms of c and l? If you are not confident in your algebra just check it against a computational engine like Wolfram Alpha, Mathematica, Maple, or whatever.
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First glance: Too lazy to do the partial differentials.
Max U(c,l) = ln(c)+Bln(l) st P(1+tc) = w(1-t)(24-l)-T
Lagrangian: L = ln(c)+Bln(l) + lambda ( P(1+tc) = w(1-t)(24-l)-T)
First order conditions (FOC):
(1) of Lagrangian w.r.t. consumption
(2) of Lagrangian w.r.t. leisure
(2) of Lagrangian w.r.t. lambda
Eliminate lambda from FOC.
Find consumption or leisure in the equation above once lambda is eliminated, then plug this into the constraint and solve for the other good (so if you isolated consumption then solve for leisure).
Then once that is found, plug that back into the good you find after FOC( so if you isolated consumption and found leisure, then plug leisure back into consumption equation).
Optimized C*, L* have been found.
Can you show me the work you have done? I'm about to go on a run but after it I am willing to review your work - conditioned on that I am on track as to what you are looking for, haven't done this in awhile!
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