Isoquant Help Please!!!?

Question

Prunella raises peaches. When L is the number of units of labor she uses and T is the number of units

of land she uses, her output is f(L, T)=L1/2T1/2 bushes of peaches.

On the graph, plot some input combinations that give her an output of 4 bushels. L is on the horizontal axis and T is on the vertical axis. Sketch a production isoquant that runs through these points.

What is the equation for the isoquant that gives her an output of 4 bushels?

The answer to this is T=16/L, but I don't understand where did the 16 came from. Please help!!!

And doesn't this isoquant function exhibit increasing returns to scale? The answer is "constant returns to scale" but I thought that as input (L) increases, if capital (in this case, T) increased as well, then the function exhibited increasing returns to scale. Why is this not the case?

Answer 1

L^1/2 T^1/2 = 4 the quantity of output

then T^1/2=4 / L^1/2

square both sides  T=16/L

Constant returns to scale means that  if you multiply both inputs by a const  value leaving the ratio unchanged  your output will increase by the same amount.  

(nL)^1/2  (nT)^1/2 =n (L^1/2 T^1/2 )

Answer 2

Isoquant means that production function gives equal amount, in your case it should give you 4 units of output at any combination of inputs.

f(L,T)=4

L1*T1/(2*2)=4

L1*T1=4*2*2=16

T1=16/L1

L1=16/T1

or how you wrote it:

T=16/L

L=16/T

Answer 3

Q = f(L, T) = L^1/2 * T^1/2

                  = (L*T)^1/2

if Q = 4,

then (LT)^1/2 = 4

square both sides,

LT = 16

divide both sides by L

T = 16/L

Done.

Viz returns to scale, keep in mind it's % changes in inputs and output that are important.  Of course Q is always going to go up when you increase input.  But returns to scale is defined as

(% change in output) / (% change in all inputs).

If that expression is less than 1, you have diminishing RTS.  Greater than 1, you have increasing RTS.  Equal to 1, you have constant RTS.

A quick way is just subbing in the relevant values for L and T, calculating Q, and then doubling L and T and seeing what happens to Q.  If Q also doubles, constant RTS.  If Q is less than double, diminishing RTS.  If Q is greater than double, increasing RTS.

There's also a shortcut for this example since it's a Cobb-Douglas function.  The rule is to add the exponents of L and T together. If the result is less than 1, then you have decreasing RTS.  If greater than 1, you have increasing RTS.  If equal to 1, you have constant RTS.

1/2 + 1/2 = 1

Thus, constant RTS.