Lachlan has the following utility function for goods x and y:
The market prices for the two goods are Px and Py, and are finite. x and y are
divisible (e.g. it is possible to purchase 0.0001 units of x).
a) Write the Lagrangian for Lachlan's utility maximisation problem.
b) Find equations for Lachlan's optimal consumption of x and y. What are
these equations called?
c) Identify the only reason why Lachlan would choose x = 0.
d) What is Lachlan's indirect utility function?
e) Comment on how changes in income and prices affect Lachlan's
happiness. Does this make sense?
f) For either x or y, check that Roy's identity works.
g) Efficiently switch from Lachlan's indirect utility function to Lachlan's
h) Use Shephard's lemma to find Lachlan's Hicksian demand for x.
i) Is x an inferior good? Explain.
j) Use the dual approach to check the solution you got for (h).