🔬 Tutorial problems beta \beta

🔬 Tutorial problems beta \(\beta\)#

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.

\(\beta\).1#

Compute the following limits:

  1. \(\quad \lim_{n \to \infty} \frac{1}{n}\)

  2. \(\quad \lim_{n \to \infty} \frac{n+2}{2n+1}\)

  3. \(\quad \lim_{n \to \infty} \frac{2n^2(n-2)}{(1-3n)(2+n^2)}\)

  4. \(\quad \lim_{n \to \infty} \frac{(n+1)!}{n! - (n+1)!}\)

  5. \(\quad \lim_{n \to \infty} \sqrt{\frac{9+n^2}{4n^2}}\)

Fact

  1. \(x_n \to a\) in \(\mathbb{R}^N\) if and only if \(\|x_n - a\| \to 0\) in \(\mathbb{R}\)

  2. If \(x_n \to x\) and \(y_n \to y\) then \(x_n + y_n \to x + y\)

  3. If \(x_n \to x\) and \(\alpha \in \mathbb{R}\) then \(\alpha x_n \to \alpha x\)

  4. If \(x_n \to x\) and \(y_n \to y\) then \(x_n y_n \to xy\)

  5. If \(x_n \to x\) and \(y_n \to y\) then \(x_n / y_n \to x/y\), provided \(y_n \ne 0\), \(y \ne 0\)

  6. If \(x_n \to x\) then \(x_n^p \to x^p\)

\(\beta\).2#

Show that the Cobb-Douglas production function \(f(k,l) = k^\alpha l^\beta\) from \([0,\infty) \times [0,\infty)\) to \(\mathbb{R}\) is continuous everywhere in its domain.

You can use the fact that, for any \(a \in \mathbb{R}\) the function \(g(x) = x^a\) is continuous at any \(x \in [0,\infty)\).

Also, remember that norm convergence implies element by element convergence.

\(\beta\).3#

Let \(A\) be the set of all consumption pairs \((c_1,c_2)\) such that \(c_1 \ge 0\), \(c_2 \ge 0\) and \(p_1 c_1 + p_2 c_2 \le M\) Here \(p_1\), \(p_2\) and \(M\) are positive constants. Show that \(A\) is a closed subset of \(\mathbb{R}^2\).

Weak inequalities are preserved under limits:

If \(x_n \leq y_n\) for all \(n\) then \(\lim_{n \to \infty} x_n \leq \lim_{n \to \infty} y_n\), including the case of constant sequence \(y_n=a\) for all \(n\).

\(\beta\).4#

Let \(f \colon [-1, 1] \to \mathbb{R}\) be defined by \(f(x) = 1 - |x|\), where \(|x|\) is the absolute value of \(x\).

  • Is the point \(x = 0\) a maximizer of \(f\) on \([-1, 1]\)?

  • Is it a unique maximizer?

  • Is it an interior maximizer?

  • Is it stationary?

Draw a graph of function \(f\).

\(\beta\).5#

Consider function \(f \colon X \to \mathbb{R}\) defined by \(f(x) = \frac{1}{x} e^x\).

Find the minimizer(s) and the maximizer(s) of this function on \(X = (0, 2]\).

Follow all the required steps and explain your reasoning.

Review the algorithm for univariate optimization in the lecture notes

\(\beta\).6#

Find an example of a nonlinear univariate function \(f \colon D \subset \mathbb{R} \to \mathbb{R}\) that:

  • (a) has exactly one maximizer and one minimizer

  • (b) has has neither a maximizer nor a minimizer

  • (c) has an infinite number of maximizers and minimizers

  • (d) has exactly finite number \(n\) of maximizers and \(n\) minimizers

Remember to define both the function \(f(x)\) and its domain \(D\) for each case.

First, review the relevant definitions. Then, try to draft some ideas on a piece of paper. Think of how they can be expressed in mathematical terms.

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