Question

Question: Solve the utility maximization problem

Max (c , l) 5c - l

subject to pc <= wl

0 <= l <= 40

0 <= c

Answer 1

Let's write the Lagrangian for the problem:

First order equations give: (If lagrangian is binding)

Dividing 1 and 2, we have 5w=p

Putting this in 3, we have wl=5wc, or l=5c

Now note that 0<=l<=40

using this 0<=5c<=40, gives 0<=c<=8

Now note that the function is 5c-l, which is = 0 for all c and l as 5c=l. So the maximum value it takes = 0 and for that 0<=c<=8, l=5c

If the constraint is not binding, then and wl>pc. i.e. l>(p/w)c and l>=0 and l<=40 and c>=0

But this would mean that the consumer is not consuming all the wealth that he/she has which is irrational and in contradiction to Walras' law in Economics