The Lagrange Method

The recipe

A weighted Cobb-Douglas worked out

The general Cobb-Douglas demands

Reading lambda as the value of money

At a Lagrange optimum, what does the multiplier lambda equal and mean?

In the example u equal to x1 x2 squared, m 90, prices 3 and 1, tangency gives x2 equals 6 x1. What does the budget then produce?

For Cobb-Douglas utility with weights a and b, what fraction of income goes to good one?

What does the Cobb-Douglas demand x1 equals a over a plus b times m over p1 say about the effect of p2 on good one?

For u equal to x1 x2 squared with income 90 and prices 3 and 1, compute x1 at the optimum, and write the number.

Same consumer, compute x2 at the optimum, and write the number.

For u of x1 times x2 squared with income 90 and prices 3 and 1, the optimum is (10, 60). What utility does it deliver?

Under the same weighted utility, how many dollars flow to each good at the optimum?

At the optimum (10, 60), lambda computed from either first-order condition gives what value?

For Cobb-Douglas demands, if p1 doubles while income and p2 hold, what happens to x1 star?

For u of x1 times x2 squared with income 90 and prices 3 and 1, compute lambda at the optimum, and write the number.

Same consumer, if the price of good two rises to 2, compute the new optimal x2 from the share law, and write the number.