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# Envelope Theorem

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Consider the problem of maximizing u(c,l) subject to pc + wl = wT + Y, where c is consumption, l is leisure time, T is the total time endowment, and Y is non-wage income. Show that if leisure is an inferior good, then the labor supply function is upward-sloping.

b) Given the problem of maximizing ln x subject to &#945; &#8805; x2, when &#945; > 0, confirm that the envelope theorem holds.

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#### Solution Preview

Problem 1

Consider the problem of maximizing u(c,l) subject to pc + wl = wT + Y, where c is consumption, l is leisure time, T is the total time endowment, and Y is non-wage income. Show that if leisure is an inferior good, then the labour supply function is upward-sloping.

*Since we want to max U(c, l) subject to pc + wl = wT + Y
We can write the Lagrangian Function as:
L = U(c, l) - m (pc + wl - wT + Y)
Where m >= 0, to be determined
Then take the first order condition:
dL/dc = dU/dc - mp = 0 or dU/dc = mp (1)
dL/dl = dU/dl - mw = 0 ...

#### Solution Summary

The attachment goes into a lot of detail related to envelope theorem. Lagrangian concepts are used to demonstrate the response to the question. Step by step instructions are provided and it is easy for any student to understand such concepts. Overall, an excellent response to the question being asked.

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