Utility function

Suppose you have a total income of I to spend on three goods x1 , x2 and x3, with unit prices p1 , p2 and p3respectively. Your taste can be represented by the utility functionu(x1,x2,x3) = x1a x2b x31-a-bwhere a and b are between 0 and 1, and a + b < 1.(a) What is your optimal choice for x1 , x2 and x3? Use the Lagrange Method.(b) What are the shares of income spent on the three goods respectively?(c) Derive your indirect utility function.(d) Derive your expenditure function.(e) If there are n goods and the utility function is: u(x1,x2,...,xn) = x1a1 x2a2 ··· xn1- a1 -a2 -...- an-1where a1, a2,..., an-1 are all between 0 and 1, and a1+a2+...+an-1 <1The unit prices are p1 , p2 ,..., pn respectively.Without calculations, write down the demand function for x1 and the share of income spent on it.

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