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  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization constrained extremum problems
  5. Method of Lagrange
  6. Exercise 1

Exercise 1

The solution of the following constrained extremum problem is given by $z(1,2)=8$.

  • Maximize $z(x,y)=xy^3$
  • Subject to $2x^2+\frac{3}{2}y^2=8$
  • Where $x,y\geq0$
Determine the corresponding shadow price.
$\lambda=2$
$\lambda=0$
$\lambda=1\frac{1}{2}$
$\lambda=8$
The solution of the following constrained extremum problem is given by $z(1,2)=8$.

  • Maximize $z(x,y)=xy^3$
  • Subject to $2x^2+\frac{3}{2}y^2=8$
  • Where $x,y\geq0$
Determine the corresponding shadow price.
Antwoord 1 correct
Correct
Antwoord 2 optie
$\lambda=0$
Antwoord 2 correct
Fout
Antwoord 3 optie
$\lambda=1\frac{1}{2}$
Antwoord 3 correct
Fout
Antwoord 4 optie
$\lambda=8$
Antwoord 4 correct
Fout
Antwoord 1 optie
$\lambda=2$
Antwoord 1 feedback
Correct: $L(x,y,\lambda)=xy^3-\lambda(2x^2+\frac{3}{2}y^2-8).$ $L'_x(x,y,\lambda)=y^3-4\lambda x$. We put these equal to zero and plug in $(x,y)=(1,2)$: $2^3-4\lambda 1=0$. Rewriting gives $\lambda=2$.

Go on.
Antwoord 2 feedback
Wrong: Use the fact that $z(1,2)=8$ is the maximum.

Try again.
Antwoord 3 feedback
Wrong: $2^3\neq 6$.

Try again.
Antwoord 4 feedback
Wrong: The shadow price is not equal to the maximum (value).

See Extra explanation: Schadow price.