Question

at the beginning of the electronic tablet era

(Only need the third question, i already posted the first 2 questions)

At the beginning of the electronic tablet era, there was only one method to produce tablets. Summarizing inputs into labor L and capital K, the proportion K/L of capital to labor use in this method (also known as capital intensity) was fixed at 2 Derive the demand for labor L(w.r.q) as a function of the price of labor w, the price of capital r and of the output level q. [Warning: this function may be constant with respect to some variables.] 1. Over time, multiple methods of production have been discovered, which substitute labor for capital in different ways, giving rise to variable capital intensity and summarized by the production function 2. Derive the demand for labor L(w.rq) as a function of the price of labor w, the price of capital r and of the output level q in the new technological environment Explain whether the tablet producing firm is more likely to reduce labor demand before or after the discovery of new methods of production, if the price of labor w increases 3.

Answer 1

3. Before the new methods of productions were discovered, its given that K/L ratio was fixed at 2. Basically, if you changed one of these, you had to change the other in such a way that the ratio remained fixed at 2. So, for every capital K I added, I had to add 2 Labour. We can write this function as

q= (L/2)^a(L)^(1-a)

OR (L)/2^a

Where a is the power of capital in the Cobb-Douglas function.

Post the discoery of new methods of production, the function has become

q= K^.25L^.75

Since after the discoery of new ways of production I am replacing capital by labour, then before the discovery the power of capital in the Cobb-Douglas function must've been higher than .25. So we get, a>.25 before the discovery. The range is .26 to .99. Lets test for both.

First, lets say it was .26. Hence if I reduce L by half before the discovery, my q becomes

L/2*2^.26 from L/2^.26

OR, it becomes

.4175440L from .835079L. A reduction of 50%.

For a=.99, q drops from .503478 to .25179, a drop of 50%.

If I reduce the labor by same after new methods, my q becomes

K^.25(L/2)^.75 from q= K^.25L^.75

OR it becomes .5946K^.25L^.75 from K^.25L^.75, a drop of 41%.

Clearly, it is better for me to reduce labor after the new methods as it will give me lesser drop in q.