2. Cost minimization with a Cobb-Douglas production function

A firm uses labor and machines to produce output according to the production function f(L,M)=4L⅟₂M⅟₂, where L is the number of units of labor used and M is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10.

a. On the graph below, use a purple line (diamond symbols) to represent an isocost line for this firm, showing combinations of machines and labor that cost $400, and then position a red line (cross symbols) for an isocost line showing combinations that cost $200. What is the slope of these isocost lines?

(Graph: Machines: (Number): 0,5,10,15,20,25,30,35,40,45,50; Labor(Units): 0,5,10,15,20,25,30,35,40,45,50))

b. Suppose that the firm wants to produce its output in the cheapest possible way. Find the number of machines it would use per worker. (Hint: The firm will produce at a point where the slope of the production isoquant equals the slope of the isocost line.)

c. On the graph above, create the production isoquant corresponding to an output of 40. Do this by plotting green points (triangle symbol) where the number of machines, M, has the following values: 2,5,10,20, and 50). Line segments will automatically connect the points. The firm should use ? of labor and ? machines to produce 40 units of output in the cheapest possible way, given the above factor prices. Calculate the cost of producing 40 units at these prices: c(40,10,40) = ? .

d. To produce y units in the cheapest possible way, the firm would use ? units of labor and machines. How much would this cost? ? (Hint: Notice that there are constant returns to scale.)

A firm uses labor and machines to produce output according to the production function f(L,M)=4L⅟₂M⅟₂, where L is the number of units of labor used and M is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10.

a. On the graph below, use a purple line (diamond symbols) to represent an isocost line for this firm, showing combinations of machines and labor that cost $400, and then position a red line (cross symbols) for an isocost line showing combinations that cost $200. What is the slope of these isocost lines?

(Graph: Machines: (Number): 0,5,10,15,20,25,30,35,40,45,50; Labor(Units): 0,5,10,15,20,25,30,35,40,45,50))

b. Suppose that the firm wants to produce its output in the cheapest possible way. Find the number of machines it would use per worker. (Hint: The firm will produce at a point where the slope of the production isoquant equals the slope of the isocost line.)

c. On the graph above, create the production isoquant corresponding to an output of 40. Do this by plotting green points (triangle symbol) where the number of machines, M, has the following values: 2,5,10,20, and 50). Line segments will automatically connect the points. The firm should use ? of labor and ? machines to produce 40 units of output in the cheapest possible way, given the above factor prices. Calculate the cost of producing 40 units at these prices: c(40,10,40) = ? .

d. To produce y units in the cheapest possible way, the firm would use ? units of labor and machines. How much would this cost? ? (Hint: Notice that there are constant returns to scale.)

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