Suppose that a farmer grows corn. The farmer's production function is Q(K,L) = 5K^0.4L^0.6, where K is the number of harvesters and L is the number of farmworker labor hours. The prevailing wage rate is $8 per hour and the rental rate associated with harvesters is $75. The farmer has a goal of delivering 100,000 bushels of corn to market. What Lagrangian equation can be used to solve the farmer's cost minimization problem?

Question

Quizplus · Accepted Answer

The Lagrangian equation in this case is: 
L = wL + rK + λ(Q - 100,000) - λ1(K - L^0.6 * (5*10^0.4)/(0.75*8*1^0.4))

where:
- w is the wage rate ($8/hr)
- r is the rental rate for harvesters ($75)
- Q is the quantity of corn produced (100,000 bushels)
- λ is the Lagrange multiplier associated with the production constraint (Q = 100,000)
- λ1 is the Lagrange multiplier associated with the cost constraint (total cost = wL + rK)

Taking the partial derivative of L with respect to L and setting it equal to 0 gives:

w - λ1*0.6*L^-0.4*(5*10^0.4)/(0.75*8*1^0.4) = 0

Taking the partial derivative of L with respect to K and setting it equal to 0 gives:

r - λ1 = 0

Solving for λ1 gives:

λ1 = r = $75

Substituting this back into the equation for L and solving for λ gives:

λ = w/0.6L^0.4*(5*10^0.4)/(0.75*8*1^0.4) = $150

Therefore, the Lagrangian equation is:

L = 8L + 75K + 150(Q - 100,000) - 150(K - L^0.6*(5*10^0.4)/(0.75*8*1^0.4))

Simplifying, we get:

L = 8L + 75K + 150Q - 15L^0.6*(5*10^0.4)/(0.75*1^0.4)

which is equivalent to option B.

Suppose That a Farmer Grows Corn