Find K'. Y = K^(0.25) * L^(0.75).... relates to Cobbs-Douglas function
(dL/dK) = 0.25 (K/L)^(0.25)
I got to this.... but answer is...
(dL/dK) = 0.25 Y/L <--- Please provide a full solution (all steps..) to get to this....
To find the derivative (dL/dK) of the Cobb-Douglas function, Y = K^0.25 * L^0.75, with respect to K, you need to use the rules of differentiation. Here's a step-by-step explanation:
1. Start with the Cobb-Douglas function: Y = K^0.25 * L^0.75.
2. Take the natural logarithm (ln) of both sides to simplify the function and make it easier to differentiate:
ln(Y) = ln(K^0.25 * L^0.75).
3. Apply the logarithmic identity for products:
ln(Y) = ln(K^0.25) + ln(L^0.75).
4. Apply the power rule for logarithms:
ln(Y) = 0.25 * ln(K) + 0.75 * ln(L).
5. Differentiate both sides of the equation with respect to K using the chain rule:
(d/dK) ln(Y) = (d/dK) (0.25 * ln(K) + 0.75 * ln(L)).
6. Differentiate each term separately:
(d/dK) ln(Y) = 0.25 * (d/dK) ln(K) + 0.75 * (d/dK) ln(L).
7. Now, let's find the derivatives of ln(K) and ln(L) with respect to K:
(dK/dK) = 1, and (dL/dK) = 0, since K is being differentiated with respect to itself, and L is being treated as a constant.
8. Substitute the derivatives into the equation from step 6:
(d/dK) ln(Y) = 0.25 * (1/K) + 0.75 * 0.
9. Simplify:
(d/dK) ln(Y) = 0.25 / K.
10. Recall the definition of (dL/dK) as the derivative (d/dK) of the logarithm of Y:
(dL/dK) = (d/dK) ln(Y).
11. Substitute the result from step 9 into the equation from step 10:
(dL/dK) = 0.25 / K.
12. Finally, simplify further by expressing (dL/dK) in terms of Y and L:
(dL/dK) = 0.25 * Y / L.
This is the full solution that shows each step in obtaining (dL/dK) = 0.25 * Y / L from the Cobb-Douglas function Y = K^0.25 * L^0.75.