A consumer has utility u(x,y,z) = ln(x) + 2ln(y) + 3ln(z) over the three goods, x,y and z and pz=1. Optimally she consumes 30 units of z. What is her income? How much money does she spend on x? (HINT: MUx = 1/x, MUy= 2/y, MUz = 3/z and remeber the "equivalent bang for the buck" condition).
(a) A consumer has utility u(x,y,z) = ln(x) + 2ln(y) + 3ln(z) over the three goods, x,y and z and pz=1. Optimally she consumes 30 units of z. What is her income? How much money does she spend on x? (HINT: MUx = 1/x, MUy= 2/y, MUz = 3/z and remeber the "equivalent bang for the buck" condition).
(b) Forget about (a). Suppose you have t= 29 hours in total to spend on 3 projects X,Y and Z to make some money.
If you spend x hours on project X, you make 2 sqrt(x) dollars;
If you spend y hours on project Y, you make 3 sqrt(y) dollars;
If you spend z hours on project Z, you make 4sqrt(z) dollars;
Writing down your "utility function" u(x,y,z) and the constraint, solve the utility maximization problem; what is the optimal amount of time to spend on x? on y? on z?
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