Answer:
a. To derive the marginal product (MP) function when K = 10, differentiate the production function with respect to L while holding K constant.
Q = L^2 + 5LK + K^2
Differentiating with respect to L:
∂Q/∂L = 2L + 5K
Substituting K = 10:
∂Q/∂L = 2L + 5(10)
∂Q/∂L = 2L + 50
So, the MP function when K = 10 is given by:
MP = 2L + 50
b. To determine the least-cost input combination to produce 45,000 units, we need to minimize the cost function C = wL + rK, where w is the price of labor (Rs. 5) and r is the price of capital (Rs. 10).
Given:
Q = 45,000
w = 5
r = 10
Using the production function:
Q = L^2 + 5LK + K^2
45,000 = L^2 + 5L(10) + 10^2
45,000 = L^2 + 50L + 100
Rewriting the equation in standard form:
L^2 + 50L - 44,900 = 0
Now, we can use the quadratic formula to solve for L:
L = (-b ± √(b^2 - 4ac))/(2a)
For a = 1, b = 50, and c = -44,900, we have:
L = (-50 ± √(50^2 - 4(1)(-44,900)))/(2(1))
L = (-50 ± √(2500 + 179,600))/2
L = (-50 ± √182,100)/2
Calculating L using the positive square root:
L = (-50 + √182,100)/2
Once we have L, we can find K using the production function:
K = (45,000 - L^2 - 5L(10))/100
The least-cost input combination for producing 45,000 units can be obtained by substituting the values of L and K into the cost function and calculating the total cost C = wL + rK.