Question

We have the production function of Ami's restaurant as below:
?(?, ?) = K^0.5 * L⁰²

a. Please determine whether this function demonstrate decreasing, increasing or constant returns to scale?

b. The restaurant currently uses 20,000 labour hours and 50 machines. Suppose that the market costs of the inputs are so stable that Ami's restaurant can use any amount of either input. If we have the use of both labour hours and machines gone up by 10%, Ami's costs will go up by precisely 10%. If we have the use of ALL inputs is gone up by 10%, will the production or the costs rise more?

c. The restaurant is able to sell as many cereals as the company produces for $20, does the profits go up by 10% with a 10% increase in input use? Explain by calculation.

Answer 1

a) ?(?, ?) = K^0.5 * L^0.2

Let's have K = aK and L = aK

thus,

new y = (aK)^0.5 * (aL)^0.2 = a^0.5K^0.5 * a^0.2L^0.2 = a^0.5+0.2K^0.5 * L^0.2 = a^0.7K^0.5 * L^0.2

new y = a^0.7  ?(?, ?)

Thus you can see that after increasing each input by a value then the output increases by by less than a. Thus it shows decreasing return to scale.

b) As you see in part (a) that the production exhibits decreasing return to scale. It implies that If it has the use of ALL inputs gone up by 10%, then the production will rise but by less than 10%.

However it is given that after increasing the inputs by 10%, the cost of production precisely increases by 10%.

Thus cost rises more than production.

c) The restaurant currently uses 20,000 labour hours and 50 machines. At these levels of units total output is -

y = K^0.5 * L^0.2 = (50)^0.5 * (20000)^0.2 = 51.249

Total Revenue = 51.249*20 = $1024.98

When inputs increase with 10%

new y = (55)^0.5 * (22000)^0.2 = 54.78

new TR = 54.78*20 = $1095.6

Thus, % change in TR = (1095.6 - 1024.98) / 1024.98 = 6.88%

As we know that by increasing the inputs by 10%, total cost increases by 10% but as you see above the total revenue increases just by 6.88%. It implies that the profit will fall as TC rises more than TR.