Question 1 [20 marks]
Explain mathematically the following statement:
"Cost minimization does not necessarily imply profit maximization," using the following information.
Constrained-cost minimization problem
The objective function is
\[
\text { Total Cost } \rightarrow T C=w \times L+r \times K
\]
and the constraint is
\[
f(L, K)=Q^{*} \quad \leftarrow \text { Prescribed output level }
\]
Profit maximization problem
\[
\operatorname{Maximize}\{\Pi=\Pi(L, K)\}
\]
where:
\[
\begin{array}{l}
\text { Total Profit } \rightarrow \Pi=\Pi(L, K)=T R-T C \\
\text { Total Revenue } \rightarrow T R=P \times Q, \\
Q=f(L, K) \quad \leftarrow \text { Production function } \\
\text { Total Cost } \rightarrow \text { TC }=w \times L+r \times K
\end{array}
\]
