(1.)  Maximize utility U = 100X - X² + 50Y - Y² + 20XY  subject to budget constraint 100 - 2X - 4Y = 0 (where the consumer's total budget is $100 and the prices of goods X and Y are $2 and $4 respectively)

1. Set up the Lagrangean and take the first partial derivatives with respect to X, Y and m (the Lagrangean multiplier on the zero-valued constraint expression). 
   
2. Set the three first partial derivatives equal to zero and solve these three equations for the utility-maximizing values of X, Y and m.
3. Check the second derivatives to make sure you are maximizing rather than minimizing utility at these values of X and Y.
4. What is the value of U at these optimal values of X and Y?
5. If the budget increases from 100 to 101, what are the optimal values of X, Y and  m?  
   What is the new level of U?  (If you have calculated this correctly, the new value of U should be very close to the old value of U plus m which implies that the Lagrangean multiplier  m represents the marginal utility of money). 
   

(2.)  In Question #2 of the previous assignment, you solved the unconstrained profit maximization problem for a firm with production function Y = K^0.4L^0.4, output price p(Y) = P = $25 and input prices p(L) = w = $4 and p(K) = r = $2.  This problem will analyze a model of the same firm trying to minimize the cost of producing a fixed level of output. 

1. First, set up the general form of the unconstrained profit maximization problem:  max Profit = PK^0.4L^0.4 - rK - wL.
   Take the first partials with respect to K and L, and set these equal to zero
   Solve these two equations algebraically for profit-maximizing input demand functions K and L.   (Here K and L are functions of P, w and r.) 
   
2. Suppose you had some data on K and L use under various levels of P, w and r, and wanted to estimate these input demand functions empirically.  If you take the logs of both sides of these input demand functions, what linear (additive) specifications of them do you get?
   
3. Are profit-maximizing K and L complements or substitutes?
4. Now, set up the "dual" cost minimization problem:
   min cost = wL + rK  subject to output constraint  Y_o = K^0.4L^0.4 where Y_o is some (positive) fixed level of output. 
   The Lagrangean formulation is min L = wL + rK + m[Y_o - K^0.4L^0.4]. 
   Take the first partials with respect to K, L and m, and set these equal to zero. 
   (Notice that combining these first-order conditions with respect to K and L and simplifying yields the same intermediate result as in the profit-maximizing case: MP_K/MP_L = r/w, where MP_K/MP_L is the slope of the isoquant curve and r/w is the slope of the isocost line.  The cost-minimizing combination of K and L can be determined for any isoquant, but there is only one combination at one isoquant that maximizes the firm's profit!)
   Solve these three equations for cost-minimimzing input demand functions K and L.  (Here K and L and are functions of Y_o, w and r.)
5. Suppose you had some data on K and L use under various levels of Y_o, w and r, and wanted to estimate these input demand functions empirically.  If you take the logs of both sides of these input demand functions, what linear (additive) specifications of them do you get?
   
6. Are cost-minimizing K and L complements or substitutes? 
   
7. Compare the profit-maximizing input demand for L against the cost-minimizing input demand for L.  Can you prove that the profit-maximizing firm's labor demand is more elastic (responsive to changes in w) than the cost-minimizing firm's labor demand?
   
8. Extending this result, which type of economy ought to respond more efficiently as input prices and consumer demands shift--a capitalist economy where firms are trying to maximize profits, or a centrally-planned economy where firms try to meet government-assigned production quotas at least cost?