🔬 Tutorial problems lambda \lambda

🔬 Tutorial problems lambda \(\lambda\)#

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.

\(\lambda\).1#

A critical point of a multivariate function is the point at which all partial derivatives are zero.

Compute the critical points of the following functions:

  1. \(\quad\) \(x^4+x^2-6xy + 3y^2\)

  2. \(\quad\) \(x^2-6xy+2y^2+10x+2y-5\)

  3. \(\quad\) \(xy^2+x^3y-xy\)

  4. \(\quad\) \(3x^4+3x^2y-y^3\)

  5. \(\quad\) \(x^2+6xy+y^2-3yz+4z^2-10x-5y-21z\)

  6. \(\quad\) \((x^2+2y^2+3z^2) e^{-(x^2+y^2+z^2)}\)

[Simon and Blume, 1994]: Exercises 17.1, 17.2

\(\lambda\).2#

A firm uses capital and labor to produce output. When it employs \(k\) units of capital and \(\ell\) units of labor, its output is \(A k^{\alpha} \ell^{\beta}\) units, where \(A\) is a positive number, and \(\alpha + \beta < 1\).

The unit price of capital is \(r\), and the unit price of labor is \(w\); both are non-negative. The firm would like to maximize the profits taking the price \(p\) of the output as given.

The firm’s chief economist Bob presented the following formulation of the firm’s optimization problem to the CEO Alice:

\[ \text{Choose } k, \ell, w \text{ and } r \text{ to maximize } p k^{\alpha} \ell^{\beta} - w \ell - r k \quad \text{ subject to } \quad \alpha + \beta < 1 \]

Questions:

  1. Is this formulation of the firm’s optimization problem correct?

  • What part reflects the revenue?

  • What part reflects the costs?

  • What are the choice variables?

  • Are there any constraints to be taken into account?

  1. Right down the problem after Alice have updated the formulation.

  2. Approach the problem as unconstrained maximization, and follow the steps in the lecture to find find all stationary points (solve the FOCs).

  3. Write down second order partial derivatives and verify the shape conditions for the profit function.

  4. What is the optimal strategy for the firm? Is the maximizer unique? Why?

\(\lambda\).3#

An airline company has regular flights between two cities, A and B. It can treat business and pleasure travelers as separate markets by demanding advance purchase and Saturday night stay0over for pleasure travelers. Suppose that it notes a demand function of \(Q=16-p\) for business travelers and \(Q=10-p\) for pleasure travelers. Suppose also that the cost function is \(C(Q)=10+Q^2\). How much should the airline charge each type of traveler to maximize their profit? Check both first and second order conditions.

\(\lambda\).4#

Imagine a consumer whose preferences are given by a utility function

\[ f(x,y) = \alpha \ln(x) + (1-\alpha) \ln(y), \; \alpha \in (0,1) \]

over two goods \(x \geqslant 0\) and \(y\geqslant 0\) faces a limited amount of one of the goods on the market. This is reflected in an additional constraint on the feasible set of consumption bungles: in addition to a standard budget constraint, consumption of good \(x\) is limited to a maximum amount of \(\bar{x}=10\).

Together with the budget constraint the feasible set is then given by

\[\begin{split} \begin{cases} x + 2 y \leqslant 16\\ x \leqslant 10 \end{cases} \end{split}\]

  • Make a sketch of the feasible set in the \(x,y\) plane. Illustrate the interior and the corner solution.

  • Solve the problem by substitution method, trying in turn each of the constraints, as well their combination, if \(\alpha=1/2\). Remember to check whether the proposed solution satisfied the left out constraint.

  • Find the set of values of preference parameter \(\alpha\) which leads to the corner solution being optimal.

First convert the inequality constraints problem to equality by noting that the utility function is strictly increasing in each of its arguments.

Then, do not forget to consider all cases when the first, the second or both constraints are binding, as illustrated here:

_images/corner_solution_1.png

Fig. 75 Interior solution#

_images/corner_solution_2.png

Fig. 76 Corner solution#

_images/corner_solution_3.png

Fig. 77 Infeasible pseudo-solution you should remember to discard#

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