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Homework answers / question archive / Consider a firm that chooses the number of workers and machines to hire
Consider a firm that chooses the number of workers and machines to hire. You are brought on as a consultant to help them minimise
costs, but have been told that they must produce y units of output. You know that with L workers and K machines, the firm can produce
(1/4)lnL+(1/2)lnK
units of output. Worker pay is w each and machines cost r each.
e) (1 point) Suppose the CEO wants to aggressively expand production,
so that the new target level is 3y. Could you simply triple your previous recommendation? Justify your answer without math.
f) (1 point) Now suppose the CEO has been reasoned with and instead
sets the new production target at y + 1. Roughly by how much will
optimised costs change? Justiify your answer without solving the optimisation problem again
We are given :
The production function :
y = (1/4)lnL+(1/2)lnK
= ln (L1/4*K1/2)
Now if we scale up the capital and labor by the factor 'c' , then we have :
F( c*L, c*K) = ln ((c*L)1/4*(c*K)1/2)
= ln c3/4 + ln (L1/4*K1/2)
F( c*L, c*K) = ln c3/4 + y ---- (1)
A) If we triple the existing capital & labor units , it means we are scaling up capital and labor by the factor 3, therefor c= 3, so when it will happen , the output would be equals to :
= ln c3/4 + y
Hence we can see that instead of trippling the output , the output increased by ln c3/4 units.
B) If he targets the new production level to be : y + 1,
Now to make it happen the , the capital & labor has to scale up by factor 'c' s.t ln c3/4 = 1 => c = e4/3
As we know that the cost function is homogeneous of degree 1 , hence the cost will also be scaled up by the factor, e4/3
