Consider the production function Q(K,L) = e^aK+bL.Replace a,b with a=9 and b=3.Assume that K>0 and L>0.Verify if the function satisfies all five Inada Conditions.
Anyone know how to solve this step by step?
To verify if the production function satisfies the Inada Conditions, we need to check five conditions individually. The Inada Conditions are as follows:
1. First Inada Condition: lim(K→0) (∂Q/∂K) > 0
2. Second Inada Condition: lim(K→∞) (∂Q/∂K) = 0
3. Third Inada Condition: lim(L→0) (∂Q/∂L) > 0
4. Fourth Inada Condition: lim(L→∞) (∂Q/∂L) = 0
5. Fifth Inada Condition: (∂²Q/∂K²) < 0 and (∂²Q/∂L²) < 0
Now, let's find out if the given production function satisfies these conditions.
Step 1: Calculate the partial derivatives of the production function Q(K, L) = e^(aK+bL) with respect to K and L.
∂Q/∂K = a*e^(aK+bL)
∂Q/∂L = b*e^(aK+bL)
∂²Q/∂K² = a²*e^(aK+bL)
∂²Q/∂L² = b²*e^(aK+bL)
Step 2: Replace the values of a and b with a = 9 and b = 3 in the partial derivatives.
∂Q/∂K = 9*e^(9K+3L)
∂Q/∂L = 3*e^(9K+3L)
∂²Q/∂K² = 81*e^(9K+3L)
∂²Q/∂L² = 9*e^(9K+3L)
Step 3: Evaluate the limit of each partial derivative as K and L approach the respective limits.
lim(K→0) (∂Q/∂K) = lim(K→0) (9*e^(9K+3L))
To evaluate this limit, we substitute K = 0 into the expression:
lim(K→0) (9*e^(9K+3L)) = 9*e^(3L)
Since e^(3L) is always greater than zero, lim(K→0) (∂Q/∂K) > 0
Now, you can follow the same steps to evaluate the remaining conditions:
lim(K→∞) (∂Q/∂K)
lim(L→0) (∂Q/∂L)
lim(L→∞) (∂Q/∂L)
(∂²Q/∂K²)
(∂²Q/∂L²)
By substituting the respective values into each condition, you can determine if the given production function satisfies all five Inada Conditions.