🔬 Tutorial problems kappa \kappa

🔬 Tutorial problems kappa \(\kappa\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\kappa\).1#

Roy’s identity

Consider the choice problem of a consumer endowed with strictly concave and differentiable utility function \(u \colon \mathbb{R}^N_{++} \to \mathbb{R}\) where \(\mathbb{R}^N_{++}\) denotes the set of vector in \(\mathbb{R}^N\) with strictly positive elements.

The budget constraint is given by \(p \cdot x \le m\) where \(p \in \mathbb{R}^N_{++}\) are prices and \(m>0\) is income.

Then the demand function \(x^\star(p,m)\) and the indirect utility function \(v(p,m)\) (value function of the problem) satisfy the equations

\[ x_i^\star(p,m) = -\frac{\partial v}{\partial p_i}(p,m) \Big/ \frac{\partial v}{\partial m}(p,m), \; \forall i \in \{1,\dots,N\} \]

  1. Prove the statement

  2. Verify the statement by direct calculation (i.e. by expressing the indirect utility and plugging its partials into the identity) using the following specification of utility

\[ u(x) = \prod_{i=1}^N x_i^{\alpha_i}, \; \alpha_i > 0 \]

Envelope theorem should be useful here.

\(\kappa\).2#

First, find the maximum and minimum square distances from the origin to the ellipse \(x^2+xy+y^2 = 3\).

Second, using the envelope theorem, approximate these square distances for the ellipse \(x^2+xy+\tfrac{9}{10}y^2 = 3\).

[Simon and Blume, 1994] Exercises 18.2 and 19.12

Use \(x^2+y^2\) as your objective function.

\(\kappa\).3#

Consider the problem of maximizing \(f(x,y)=x^2+x+4y^2\) subject to the inequality constraint

\[ 2x+2y \leq 1, \; x \geq 0, \; y \geq 0. \]

Using the envelope theorem, approximate the maximum value of \(x^2+x+4.1 y^2\) on the same constraint set.

[Simon and Blume, 1994] Exercises 19.13

As the envelope theorem requires knowing the optimizer and the corresponding Lagrange multiplier, the problem has to be first solved using KKT method.

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