You have to nd the utility maximizing consumption bundle in the following problem.

Utility function of an agent is u : R 2 + R, with u(x, y) = x y .

The per unit prices of goods x and y are px and py dollars, respectively.

The agent can spend at most M dollars on purchasing a consumption bundle.

> 0, > 0, px > 0, py > 0, M > 0

[1] What are the "choice variables" and what are the "parameters" in this problem? What is the dierence between them?

[2] How do we know this problem has a solution?

[3] Explain the relevant Lagrangian function.

[4] Explain the conditions that will characterize a KT-point.

[5] Use the conditions in part [4] to nd the optimal consumption bundle (x , y ), and the associated Lagrange multiplier .

[6] Explain why x and y can be interpreted as "functions".

[7] Find the partial derivative of x with respect to each "parameter of the problem".

[8] Explain the interpretation of each partial derivate you nd in part [7].

[9] Find the "indirect utility function" V = u(x , y ), and explain its interpretation.

[10] Find the partial derivative of V with respect to M, and explain its interpretation.