Question

1. Consider the following Cobb-Douglas Production Function for Mauricio’s Machines Inc, a U.S. manufacturer of electronic precision tools:
Q = (L * 0.4) + (K * 0.7)
a. Explain why a Cobb-Douglas production is indeed a representation of reality?

b. Based on the function above, is Mauricio’s Machines’ business experiencing Economies of Scale or Diseconomies of Scale? Please explain your answer.

c. If Mauricio’s Machines decided to boost Labor by 10% and Capital by 15%, how much would their productivity increase?

d. If Mauricio’s Machines wanted to boost productivity by 50% and already knew they were going to increase capital by 25%, how much would they have to increase their labor force to reach this production target?

Answer 1

1. Cobb-Douglas function represents reality because it takes 3 inputs which make the basis of all real production- labor, capital and productivity. Any prodcution requires labor (that is, effort from humans) and capital (money which buys everything else). Finally, the output also depends of efficiency- how efficient the labor is, the technology is and so on. This is what A in the function represents.

Finally, it also has the two values of alpha and beta (the power), these represent how the production changes as more (or less) labor and capital is added. These are elasticities of inputs.

In light of above, its easy to see how the Cobb Douglas function indeed represents reality.

2. When alpha+beta>1, meaning the firm is experiencing economies of scale. Since alpha and beta represent the elasticcities of inputs, alpha+beta>1 means dobling both input factor will more than double the output. As output is increasing faster than inputs, its economies of scale.

In our case, alpha+beta is .4+.7=1.1

Hence, economies of scale.

3. The current production is

Q=L^.4K^.7

The new production function would be

Q=(1.1L)^.4(1.15K)^.7

Q=1.146L^.4K^.7

Hence, the new production is 14.6% higher

4. Lets say they need to increase their labor productivity by x.

The new production needs to be

1.5L^.4K^.7=((1+x)L)^.4(1.25K)^.7

1.5L^.4K^.7=1.1691*(1+x)^.4L^.4K^.7

1.5=1.1691*(1+x)^.4

Solving for X, we get

x=.865

Meaning, the labor force would need to be increased by 86.5%.