Consider the following Production Functions: F(K,L)=AKLF(K,L)=AKLF(K,L)=KL(1)F(K,L)=K+LF(K,L)=K21+L21 And follow the steps shown in the Example. Write the cost-minimization problem as: MinimizeC=wL+rK Subject to the constraint: F(K,L)=q0 Step 1: Set up the Lagrangian, which is the sum of two components: the cost of production to be minimized and the Lagrange multiplier times the output constraint faced by the firm: =wL+rK[F(K,L)q0] Step 2: Differentiate the Lagrangian with respect to K,L, and . Then equate the resulting derivatives to zero to obtain the necessary conditions to a minimum. L=0K=0=0 Example: Minimize C=wL+rK Subject to the constraint that a fixed output q0 be produced: F(K,L)=AKL=q0 Step 1: Set up the Lagrangian, which is the sum of two components: the cost of production to be minimized and the Lagrange multiplier times the output constraint faced by the firm: =wL+rK[AKLq0] Step 2: Differentiate the Lagrangian with respect to K,L, and . Then equate the resulting derivatives to zero to obtain the necessary conditions to a minimum. K=rKF(K,L)=r(K1L)=0L=wLF(K,L)=w(AKL1)=0=AKLq0=0 Consider the following Production Functions: F(K,L)=AKLF(K,L)=AKLF(K,L)=KL(1)F(K,L)=K+LF(K,L)=K21+L21 And follow the steps shown in the Example. Write the cost-minimization problem as: MinimizeC=wL+rK Subject to the constraint: F(K,L)=q0 Step 1: Set up the Lagrangian, which is the sum of two components: the cost of production to be minimized and the Lagrange multiplier times the output constraint faced by the firm: =wL+rK[F(K,L)q0] Step 2: Differentiate the Lagrangian with respect to K,L, and . Then equate the resulting derivatives to zero to obtain the necessary conditions to a minimum. L=0K=0=0 Example: Minimize C=wL+rK Subject to the constraint that a fixed output q0 be produced: F(K,L)=AKL=q0 Step 1: Set up the Lagrangian, which is the sum of two components: the cost of production to be minimized and the Lagrange multiplier times the output constraint faced by the firm: =wL+rK[AKLq0] Step 2: Differentiate the Lagrangian with respect to K,L, and . Then equate the resulting derivatives to zero to obtain the necessary conditions to a minimum. K=rKF(K,L)=r(K1L)=0L=wLF(K,L)=w(AKL1)=0=AKLq0=0