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  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization constrained extremum problems
  5. Method of substitution

Method of substitution

Introduction: A constrained extremum problem is given by
  • Optimize $z(x,y)$
  • Subject to $g(x,y)=k$
  • Where $x \in D_1$, $y \in D_2$
Method:
  1. Rewrite $g(x,y)=k$ as a function $y(x)$.
  2. Replace $y$ in $z(x,y)$ by $y(x)$: $Z(x)=z(x,y(x))$.
  3. Optimize $Z(x)$ as a function of one variable. This gives extremum location $c$.
  4. Since $Z(c)=z(c,d)$, with $d=y(c)$, it holds that $z(c,d)$ is the extremum of $z(x,y)$.
Remark: The substitution cannot be used if $g(x,y)=k$ cannot be written as a function $y(x)$ (or $x(y)$).