Lachlan has the following utility function for goods x and y:

????(????,????)=9????^{0.2}????^{0.6}+10

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

expenditure function.

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).