1. Make up your own constrained optimization problem for an objective function with two independent...

Question

1. Make up your own constrained optimization problem for an objective function with two independent variables and solve it using the substitution method.

2. A firm faces the production function Q = 2K^0.2L^0.6 and can buy L at £240 a unit and K at £4 a unit.

(a) If it has a budget of £16,000 what combination of K and L should it use to maximize output?

(b) If it is given a target output of 40 units of Q what combination of K and L should it use to minimize the cost of this output?

David · Accepted Answer

1. A firm’s production function is given by f(K,L) = K2 + L2 . We want to find the expansion path 
of the firm and its total cost function as a function of X. 
To determine the expansion path we maximise output x subject to the cost constraint rK + 
wL = c. For this let us consider the Lagrangian expression,
                               L  =  K2 + L2 – λ (rK + wL -c)
where λ is the Lagrangian multiplier. The first order conditions for the constrained 
maximisation problem are 
                            δL/ δ K = 2K – λr = 0
                            δL/ δ L = 2L – λw = 0
                            δL/ δ λ = -rK -wL + c = 0
Using the first two conditions we get 
                                       K/L = r/w                                                                                     (1)
Thus, with r and w given, the right hand side is a constant. If we denote r/w by b, then the 
above equation becomes K = bL, which is a straight line expansion path through origin.
To determine the total cost as a function of output we need to consider the problem of 
minimisation of cost for each level of output. We have noted that the dual to this primal cost 
minimisation is:

1. Make up your own constrained optimization problem for an objective function with two independent variables and solve it using the substitution method. 2. A firm faces the production function Q = 2K...

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