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Question 6. An aircraft manufacturing firm can operate plants in either of two countries. In country A, its cost as a function of output $x \ge 0$ is $C_A(x) = \ln(1 + 3x/100)$. In country B, its cost as a function of output $y \ge 0$ is $C_B(y) = 2\ln(1 + y/100)$. The firm allocates production between the two plants in order to minimize the total cost of producing at least q units of output worldwide, where $q > 0$. (a) Show that the firm's cost-minimizing choices of $x$ and $y$ must solve a particular constrained optimization problem with non-negativity constraints. (b) Use the Lagrange multiplier method to show that there are one, two, or three solution candidates satisfying the Kuhn-Tucker conditions, depending on the value of $q$.

          Question 6. An aircraft manufacturing firm can operate plants in either of two countries.
In country A, its cost as a function of output $x \ge 0$ is $C_A(x) = \ln(1 + 3x/100)$. In country
B, its cost as a function of output $y \ge 0$ is $C_B(y) = 2\ln(1 + y/100)$. The firm allocates
production between the two plants in order to minimize the total cost of producing at least
q units of output worldwide, where $q > 0$.
(a) Show that the firm's cost-minimizing choices of $x$ and $y$ must solve a particular constrained
optimization problem with non-negativity constraints.
(b) Use the Lagrange multiplier method to show that there are one, two, or three solution
candidates satisfying the Kuhn-Tucker conditions, depending on the value of $q$.

Question 6. An aircraft manufacturing firm can operate plants in either of two countries.
In country A, its cost as a function of output x ≥ 0 is CA(x) = ln(1 + 3x/100). In country
B, its cost as a function of output y ≥ 0 is CB(y) = 2ln(1 + y/100). The firm allocates
production between the two plants in order to minimize the total cost of producing at least
q units of output worldwide, where q > 0.
(a) Show that the firm's cost-minimizing choices of x and y must solve a particular constrained
optimization problem with non-negativity constraints.
(b) Use the Lagrange multiplier method to show that there are one, two, or three solution
candidates satisfying the Kuhn-Tucker conditions, depending on the value of q.

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