Cobb-Douglas preferences

A consumer has the following preferences

u(x₁ , x₂) = log (x₁) + log (x₂)

Suppose the price of good 1 is p₁ and the price of good 2 is p₂. The consumer has

income m.

(a) Write down optimal choices for the utility maximization problem.

(b) What is the own-price elasticity of demand for good 1? What is the cross-price

elasticity of demand for good 1?

(c) What is the income elasticity of demand for good 1? Plot the Engel curve for

good 1.

(d) Solve the expenditure minimization problem and find the optimal solution.

(e) Find compensating variation from change in p₂ from a to b using a method of

your choosing (a < b). Is this the same as consumer surplus lost due to this change?

[Hint: You don't need to calculate CS.]