What is the cost of producing 100 units of output for a firm with a production function given by F(k,l)=10(min(2k,l))^0.5? Assume that r=20 $/unit and w=10 $/unit

To determine the cost of producing 100 units of output for the given production function, we need to calculate the quantities of capital (k) and labor (l) required to achieve this level of production.

First, let's denote the cost of capital (r) as $20 per unit and the cost of labor (w) as $10 per unit.

The production function is given as F(k, l) = 10 * (min(2k, l))^0.5

To produce 100 units of output, we need to find the values of k and l that satisfy this condition.

We can start by assuming different values for k and calculating the corresponding value of l using the production function. Then we can check if the calculated output is equal to or exceeds 100 units.

For example, let's assume k = 10. Plugging this value into the production function gives:

F(10, l) = 10 * (min(2*10, l))^0.5
= 10 * (min(20, l))^0.5

Now, we can solve for l by equating the output to 100 units:

100 = 10 * (min(20, l))^0.5

To isolate the min(20, l) term, we can square both sides of the equation:

(100/10)^2 = (min(20, l))^0.5

Simplifying further:

10^2 = min(20, l)

Since the minimum of 20 and any value of l will be 20, it means that we can set l = 20.

Therefore, if k = 10 and l = 20, the production function will yield an output of 100 units.

Now that we know the values of k and l, we can calculate the cost of producing 100 units of output.

The cost of capital (r) is $20 per unit, so the cost of capital (Ck) would be:

Ck = r * k
= 20 * 10
= $200

Similarly, the cost of labor (w) is $10 per unit, so the cost of labor (Cl) would be:

Cl = w * l
= 10 * 20
= $200

To get the total cost (C) of producing 100 units of output, we sum up the cost of capital and labor:

C = Ck + Cl
= $200 + $200
= $400

Therefore, the cost of producing 100 units of output for this firm would be $400.