Question

6. A firm’s production function is Y = K½ L½ [Note: This is the same thing as saying the “square root” of K*L. In other words if K = 9 and L = 9, it could produce 3*3 (or alternatively the square root of 81) = 9 units of output.]

a) Draw the isoquants corresponding to 1, 2, 3, and 4 units of output produced. (This may require some thinking… to get you started, for the 2 unit isoquant, set Y = 2, so K½ L½. =2. Square both sides and now K*L = 4. K = 4/L. From here you should be able to come up with a few points to draw on the isoquant and extrapolate its shape.) (2 pts)

b) Suppose that the firm produces 10 units of output using 20 units of capital and 5 units of labor. Compute the MPL by allowing labor to rise by one unit to 6 and comparing the old and new output level. Do the same for MPK. (2 pts)

c) Compute the MRTS based on your answer to (b) (1 pt)

d) This is a 2 point BONUS Question: Of course your answers to (b) and (c) above are imperfect approximations. Now try to compute the MPL and MPK using calculus… the MPL is ∂Y/∂L, which is ½* K½ L-½, which is the same thing as saying “one half times the square root of (K/L).” Compute the MRTS at 20 units of K and 5 units of L producing 10 units. (If you are one of the students who has yet to take calculus, feel free to ask someone for help or consult my “crash course” handout if you want to try for these bonus points.)

e) Assuming this firm is optimizing by using 20 units of capital and 5 units of labor to produce 10 units of output, what must the price ratio between labor and capital (w/r) be? (If you calculated D for bonus points, use this MRTS, if you did not, use MRTS from C). (1 pt)

Answer 1

a).

Consider the following table and fig of isoquant.

b).

Consider the given problem here the production function is given by, “Y = K^0.5*L^0.5”. So, if “K=20” and “L=5”, the “Y=(K*L)^0.5 = 100^0.5=10. Now, as “L” increases by 1, to “L=6”, the corresponding output is , “Y=(K*L)^0.5 = (20*6)^0.5 = 120^0.5 = 10.95. So, the MPL, is the difference between “10.95” and “10”. SO, here the “MPL” is “0.95”.

Now, as “K” increases by 1, to “K=21”, the corresponding output is , “Y=(K*L)^0.5 = (21*5)^0.5 = 105^0.5 = 10.25 So, the MPK, is the difference between “10.25” and “10”. SO, here the “MPK” is “0.25”.

c).

As we know that the MRTS is the ratio of “MPL” and “MPK”. SO, here the MRTS, is “MPL/MPK=0.95/0.25=3.8.

d).

The production function is given by, “Y = K^0.5*L^0.5”, => MPL = (1/2)*(K/L)^0.5, => at “K=20” and “L=5”, “MPL = 0.5*(20/5)^0.5 = 0.5*4^0.5=0.5*2 = 1 > 0.95.

Similarly, MPK = (1/2)*(L/K)^0.5, => at “K=20” and “L=5”, “MPK = 0.5*(5/20)^0.5 = 0.5*(1/4)^0.5=0.5*0.5 = 1/4=0.25.

So, the MRTS is the ratio of MPL to MPK, => MRTS is “MPL/MPK”, => (1/1/4) = 4 > 3.8.

e).

The firm is optimizing by producing 10 units, where “K=20” and “L=5”. So, here the “W/R”, must be equal to “MRTS”. So, the input price ratio is “MRTS=4”.