Question

For Firm B robots and engineers are perfect substitutes.Each robot can work K hours on its own. Alterna ely, each engineer can work L hours on her own. Because robots are not subject to fatigue, a robot produces twice the amount an engineer produces in an hour ii. Which of the following three production functions captures the fact that "a robot produces twice the amount an engineer produces in an hour"? Explain (in a few lines) why you believe your choice is correct. iv. Suppose that firm B wants to produce Q 12 units. For each combination of the cost of inputs below, find the optimal number of hours (K and L) robots and/or humans should work to produce Q 12 units. (Said differently, find K and L which guarantee Q=12 at the lowest cost? Cost of inputs r = 2€ w 2 w1.50 Optimal K (Robot hours) Optimal L (Human hours)

Answer 1

iii Since productivity of robots is twice the productivity of engineers, the marginal product of robots will be twice the margnal product of engineers. Hence, $MPK=2MPL$

The third production function displays this property.

Q=2K + L

iv The firm's cost minimization problem is to minimize cost, given by subject to
Q=2K + L

Since the firm treats robots and engineers as perfect substitutes, it would only employ the cheaper of the two. Therefore, the combinations of engineers and robots can be given by:

(K. L) = ǐio.2?), for r > 2w

In the first case, $r<2w \rightarrow 2<3$ Hence,

12 (K, L)-(ה,01-16,0 )

In the second case, $r<2w\rightarrow 2<4$ Hence,
12 (K, L)-(ה,01-16,0 )

In the third case, $r>2w\rightarrow 5>4$ Hence,

12 2 (K, L) (0,(0, 12)

To better understand how we got to these results, let's plot the production function and the cost function

0 8 6 4 2 0 8 4 2 2 6 siaaulau3'o Apueno

The blue line represents the firm's production function which is constrained at 12 units of production. The red line is the firms cost function. The firm's objective is to minimize costs. Now, as long as the slope of the red line is less than the slope of the blue line, the firm will always choose to employ only robots for production. When the slope of the red line is greater than the slope of the blue line (red line is steeper than the blue line), the firm will only employ engineers. If the slope of the two lines are equal, the firm will be indifferent between employing the two factors and would go for any combination that produces the given level of production.

In this case, the slope of the blue line is 2 and the slope of the red line is given by the ratio of the factor prices of robots and engineers. Hence, if

- 2 or r2w

the firm will employ only engineers.