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# Short run and long run cost minimization problem

I am having problems trying to solve the following problem. Please provide me with detailed step-by-step solutions.

Suppose that a firm's production function is Q = 10 L 0.5 K 0.5
The cost of a unit of L is 20\$ and user cost of capital is 80\$.

a) The firm is currently producing 100 units of output and has determined that the cost-minimizing quantities (short and long run) of L and K are 20 and 5 respectively. Graphically illustrate this using an isoquant and isocost. What is the slope of the isocost at this point of cost-minimization?

b) The firm now wants to increase output to 140 units in the short run. Illustrate this graphically and find the firm's new total cost. Make sure to show the short run expansion path.

c) Find the cost minimizing level of K and L required to produce 140 units of output. What is the total cost of 140 units in the long run? Graphically identify the cost minimizing level of K and L in the long run if the firm wants to produce 140 units of output. Make sure to show the long run expansion path.

See attached file for full problem description.

#### Solution Summary

Short run and long run cost minimization problems are solved.

\$2.19