3 Production (6 points) The firm Avanti produces custom-made sports cars that ap
ID: 1146840 • Letter: 3
Question
3 Production (6 points) The firm Avanti produces custom-made sports cars that appeal to a certain clientele because of their high price and thus exclusivity. Avanti has the following production function: q = f(k, l) = 4k2/311/3 where k and 1 are the capital and labor input respectively. (a) What additional output does one additional (small) unit of capital produce when the capital- (b) If you employed twelve workers less (again, small units), how many additional units of capital labor ratio is (k/l)- 8? (2) would you need to produce the same output? Assume that the capital-labor ratio is (k/1-8. Do not assume this any further.] (2) (c) How many units of labor does the firm need to produce 80 units of output in the short run when the capital stock is fixed at a level of k = 5? (2)Explanation / Answer
Consider the given problem, here there are 2 inputs with production function given below.
=> q = 4*k^(2/3)*l^(1/3).
a).
The “marginal product of k” is given by, “MPk = q/k = 4*l^(1/3)*(2/3)*k^(-1/3).
=> MPk = 4*l^(1/3)*(2/3)*k^(-1/3) = (8/3)*l^(1/3)*k^(-1/3) = (8/3)*(l/k)^(1/3) = (8/3)*(1/8)^(1/3).
=> MPk = (8/3)*0.5 = 1.33.
=> if we increase “k” 1 unit, => “q” will increase by “1.33” units.
b).
So, here the production function is given by, “q = 4*k^(2/3)*l^(1/3)”.
Now, MPl = 4*k^(2/3)*(1/3)*l^(-2/3) = (4/3)*(k/l)^(2/3) = (4/3)*8^(2/3) = 16/3 = 5.33.
So, the “MPl=5.33” and “MPk=1.33” at the capital-labor ratio, “8”.
So, the MRTSlk=MPl/MPk = 5.33/1.33 = 4.0075, => dK/dL = (-4.0075) = (-4).
So, if we employed 12 worker less, => “dL = (-12)”, => dK = (-4)*(-12) = 48 > 0.
So, here we need to increase “48” unit more “K” to produce same level of “q”.
c).
Here the production function is given below.
=> q = 4*k^(2/3)*l^(1/3). So here we need to find out the value of “l”, at q=80 and k=5.
=> 80 = 4*5^(2/3)*l^(1/3), => 80/4 = 5^(2/3)*l^(1/3) = 20, => l*5^2 = 20^3, => l = 8000/25 = 320.
=> we in the SR we need “l=320” if we want to produce “q=80” at “k=5”.
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