Multivariable Optimization (Chain Rule vs. Plugging In)

[Scyzo]

I am stuck on the following optimization problem:

P(K,L,T) = p(a ln(K) +b ln(L) + c ln(T)) - rK - wL - qT

1.) Find the values for K*, L* & T* that maximize the function (assume that a maximum exists)

2.) Let P* denote the optimum as a function of r, w, and q. Compute the partial derivative dP*/dr.

My problem is with the second part.

The answer to 1.) yields the following partials:

dP/dK = pa/K -r --> K* = pa/r

dP/dL = pb/L -w --> L* = pb/w

dp/dT = pc/T - q --> T* pc/q

Now I tried to approach 2.) by applying the multivariable chain rule i.e.:

dP*/dr = dP/dK (dK*/dr) + dP/dL (dL*/dr) + dp/dT (dT*/dr)

with

dK*/dr = -pa/(r^2)

dL*/dr = 0

dT*/dr = 0

= (pa/K - r)(-pa/(r^2)) = -pa/r + pa/r = 0

This is not the official solution.

However, when I plug K* = pa/r into the original function and then calculate the partial dK*/dr I arrive at the solution dK*/dr = -pa/r which is the correct solution.

Why is this the case? i.e. why does the multivariable chain rule result in 0 in this case? I am struggling a bit to see through the logic in this case of why plugging in works but the chain rule doesn't. Any clarification would be greatly appreciated!

 

[Deleted member 4993]

I am stuck on the following optimization problem:

P(K,L,T) =p * (a ln(K) +b ln(L) + c ln(T)) - rK - wL - qT

1.) Find the values for K*, L* & T* that maximize the function (assume that a maximum exists)

2.) Let P* denote the optimum as a function of r, w, and q. Compute the partial derivative dP*/dr.

My problem is with the second part.

The answer to 1.) yields the following partials:

dP/dK = pa/K -r --> K* = pa/r

dP/dL = pb/L -w --> L* = pb/w

dp/dT = pc/T - q --> T* pc/q

Now I tried to approach 2.) by applying the multivariable chain rule i.e.:

dP*/dr = dP/dK (dK*/dr) + dP/dL (dL*/dr) + dp/dT (dT*/dr)

with

dK*/dr = -pa/(r^2)

dL*/dr = 0

dT*/dr = 0

= (pa/K - r)(-pa/(r^2)) = -pa/r + pa/r = 0

This is not the official solution.

However, when I plug K* = pa/r into the original function and then calculate the partial dK*/dr I arrive at the solution dK*/dr = -pa/r which is the correct solution.

Why is this the case? i.e. why does the multivariable chain rule result in 0 in this case? I am struggling a bit to see through the logic in this case of why plugging in works but the chain rule doesn't. Any clarification would be greatly appreciated!

This problem does not make sense!!

When you define:

P(K,L,T) =p * (a ln(K) +b ln(L) + c ln(T)) - rK - wL - qT

You are saying r, w and q are constants wrt P. In that case you CANNOT calculate dP/dr.

 

[Scyzo]

Thanks for your reply. I understand the problem as follows: In 1.) r,w & q are treated as constants. In 2.) P*(K*,L*,T*) they are variables of K*(r), L*(w), and T*(q).

Also note that P (capital) denotes the function and p (lowercase) is a constant.