Consider a firm that chooses the number of workers and machines to hire

We are given :

The production function :

y = (1/4)lnL+(1/2)lnK

= ln (L^1/4*K^1/2)

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 c^3/4 +   ln (L^1/4*K^1/2)

F( c*L, c*K) = ln c^3/4 + 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 c^3/4 + y  

Hence we can see that instead of trippling the output , the output increased by ln c^3/4 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 c^3/4 = 1 => c =  e^4/3

As we know that the cost function is homogeneous of degree 1 , hence the cost will also be scaled up by the factor, e^4/3