Utility Maximization

3. 2-good Utility Maximization (Show your steps!)

In a world of two goods x and y, a utility maximizing consumer has utility function u(x;y) = x 1 2y

1 3 : The price on x and y are respectively px = 2 and py = 1 and she has a budget of

m = 10. So the consumer is constrained by pxx+pyy = 10 when maximizing here utility u. (Suppose the consumer always spends her whole budget.) Solve for her optimal choice of x, y and the corresponding utility level.

4. General Cobb-Douglas Utility Maximization. (Show your steps!)

A Cobb-Douglas utility function of two goods is of the form u(x;y) = x�y� after solving a special case in questions 3. Solve for the general problem

Max x;y

u(x;y) = x�y�

s:t: pxx+pyy = m

Your solution x� and y� should be functions of the parameters �;�;px;py;m; where all parameters are positive.

5. 3-good Utility Maximization (Show your steps!)

Solve the 3 good utility maximization problem

Max x;y;z

u(x;y;z) = xy +z

s:t: pxx+pyy +pzz = m

Your solution x�; y� and z� should be functions of the parameters px;py;pz;m; where all parameters are positive.

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6. 2-input Cost Minimization (Show your steps!)

A �rm is bounded by its production function y = f (x1;x2) = x 1 2 1 x

1 3 2 ; where x1 and x2

are quantities of the 2 inputs used to produce its product. The �rm�s cost function is c(x1;x2) = !1x1 + !2x2; where !1 and !2 are input unit prices/costs. Suppose !1 = 2 and !2 = 1 and the �rm targeted at producing y = 10 units of products. Solve for its optimal choice of x1 and x2: