Rosenberg produces board games using Labour(L) and Machines(K) as inputs. His bo
ID: 1227367 • Letter: R
Question
Rosenberg produces board games using Labour(L) and Machines(K) as inputs. His board game production function is given as follows: Q = f(K;L) = 15K^3/4 + L^1/2 At the end of last year Rosenberg and only machine for $2,000. After using this bought his first machine for 5 years it will lose all its value. Rosenberg calculates depreciation linearly (depreciation will be 20% a year in this case). There is no other use for this machine and it will have no value after the five years are up. Rosenberg is not able to buy any more machines at this moment. (a) What type of returns to scale (increasing/constant/decreasing) does Rosenberg's production function exhibit? (b) If Rosenberg's production function exhibits increasing returns to scale how could it be transformed to decreasing returns to scale? If it exhibits constant returns to scale how could it be transformed to increasing returns to scale? If it exhibits decreasing returns to scale how can it be transformed to constant returns to scale? (c) What is Rosenberg's annual fixed cost of production? Is the fixed cost sunk or not? Explain your answer. (d) Rosenberg pays a wage equal to 3. What is Rosenberg's annual total cost function?Explanation / Answer
a) Q = 15K^0.75 + L^0.5
Now the addition of 0.75 and 0.5 is more than 1 so it is an increasing return to the scale function.
b) Here labor and machine are resources used for production function. Decreasing one of the resource will bring down the production and it could be turned into decreasing returns to scale
For example if labor is halved and machine use is reduced by 1/3rd then
Q = 15K^2/4 + L^1/4
0.5 + 0.25 = 0.75
0.75 < 1 so it is decreasing returns to scale now.
c) The cost of machine is fixed cost and it has bought for $ 2000. It's life is 5 years and depreciation is at 20% per year.
2000 * 20/100 = 400
Annual cost is $ 400 per year.
This cost is not sunk cost. Although it has been already incurred but as long as production will continue, it can recovered and hence could not termed as sunk cost.
Sunk cost is the one which has been incurred and can not be recovered.
d) Machine cost is 400 and labor cost is 3.
Putting these values in equation
Q = (15 * 400^0.75) + ( 3^0.5 )
= 1343.37
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