A firm uses labor and capital as factors for a technology with constant returns to scale and strictly convex production function. A new manager, looking through past data, observed that:
when factor prices were: wl = 20 and wk = 5; the firm hired l = 10 and k = 30 to produce output
y = 100;
? when factor prices were: wl = 5 and wk = 12; the firm hired l = 18 and k = 36 to produce output
y = 150:

Assume that the past choices of labor and capital minimized costs given respective price levels.

a. Suppose that the current prices of factors are equal to wl = 5 and wk = 12: How much labor and capital should be hired if the new manager wants to produce output 300? How much if he wants to produce outoput 100? (Hint: Use the fact that the technology has a constant returns to scale).

Next, suppose that the current prices of factors change to wl = 10 and wk = 10 and the firm needs to hire labor and capital in order to produce y = 100 output: The deputy manager comes up with the following propositions:

b. l = 20; k = 10;
c. l = 12; k = 30;
d. l = 11; k = 23;
e. l = 11; k = 28;
f. l = 5; k = 100:

Which of the above propositions should be rejected by the manager?