(a) To find the value of Q when K = 81 and L = 32, we substitute these values into the production function:

Q = 100K^(3/4)L^(1/5)
Q = 100*(81)^(3/4)*(32)^(1/5)

First, calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 81^(3/4) = 27
L^(1/5) = 32^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*27*2 = 5400

(b) If we halve the labour and capital, K becomes 81/2 = 40.5 and L becomes 32/2 = 16. Substituting these values into the production function:

Q = 100*(40.5)^(3/4)*(16)^(1/5)

Calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 40.5^(3/4) = 15.59 (approx)
L^(1/5) = 16^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*15.59*2 = 3118 (approx)

So, the value of Q decreases when we halve the labour and capital.

(c) To find an expression for ln Q in terms of ln K and ln L, we take the natural logarithm of both sides of the production function:

ln Q = ln [100K^(3/4)L^(1/5)]

Using the properties of logarithms, this simplifies to:

ln Q = ln 100 + (3/4)ln K + (1/5)ln L

Question

(a) To find the value of Q when K = 81 and L = 32, we substitute these values into the production function:

Q = 100K^(3/4)L^(1/5)
Q = 100*(81)^(3/4)*(32)^(1/5)

First, calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 81^(3/4) = 27
L^(1/5) = 32^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*27*2 = 5400

(b) If we halve the labour and capital, K becomes 81/2 = 40.5 and L becomes 32/2 = 16. Substituting these values into the production function:

Q = 100*(40.5)^(3/4)*(16)^(1/5)

Calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 40.5^(3/4) = 15.59 (approx)
L^(1/5) = 16^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*15.59*2 = 3118 (approx)

So, the value of Q decreases when we halve the labour and capital.

(c) To find an expression for ln Q in terms of ln K and ln L, we take the natural logarithm of both sides of the production function:

ln Q = ln [100K^(3/4)L^(1/5)]

Using the properties of logarithms, this simplifies to:

ln Q = ln 100 + (3/4)ln K + (1/5)ln L

Knowee AI · Accepted Answer

(a) To find the value of Q when K = 81 and L = 32, we substitute these values into the production function:

Q = 100K^(3/4)L^(1/5)
Q = 100*(81)^(3/4)*(32)^(1/5)

First, calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 81^(3/4) = 27
L^(1/5) = 32^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*27*2 = 5400

(b) If we halve the labour and capital, K becomes 81/2 = 40.5 and L becomes 32/2 = 16. Substituting these values into the production function:

Q = 100*(40.5)^(3/4)*(16)^(1/5)

Calculate the values of K^(3/4) and L^(1/5):

K^(3/4) = 40.5^(3/4) = 15.59 (approx)
L^(1/5) = 16^(1/5) = 2

Then substitute these values back into the equation:

Q = 100*15.59*2 = 3118 (approx)

So, the value of Q decreases when we halve the labour and capital.

(c) To find an expression for ln Q in terms of ln K and ln L, we take the natural logarithm of both sides of the production function:

ln Q = ln [100K^(3/4)L^(1/5)]

Using the properties of logarithms, this simplifies to:

ln Q = ln 100 + (3/4)ln K + (1/5)ln L

Consider the production functionQ = 100K3/4L1/5where Q is the quantity of goods produced, K is the capital and L is thelabour.(a) Find the value of Q when K = 81 and L = 32.(b) What happens to the value of Q if we halve the labour and capital?(c) Find an expression for ln Q in terms of ln K and ln L