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🔬 Tutorial problems delta

🔬 Tutorial problems delta#

\(\delta\).1#

Consider two convergent sequences in \(\mathbb{R}^n\), \(\{{\bf x}_i\}_{i=1}^\infty\) and \(\{{\bf y}_i\}_{i=1}^\infty\) such that

\[ \lim_{i \to \infty} {\bf x}_i = {\bf x} \in \mathbb{R}^n, \quad \lim_{i \to \infty} {\bf y}_i = {\bf y} \in \mathbb{R}^n \]

Prove the following properties of the limits:

  • \(\lim_{i \to \infty} ({\bf x}_i+{\bf y}_i) = {\bf x} + {\bf y}\)

  • \(\lim_{i \to \infty} ({\bf x}_i'{\bf y}_i) = {\bf x}'{\bf y}\)

  • \({\bf x}_i \le {\bf y}_i\) for \(\forall i\) component-wise \(\implies {\bf x} \le {\bf y}\)

Definition

The scalar product \({\bf x}'{\bf y}\) of two vector is \(\mathbb{R}\) is defined by \({\bf x}'{\bf y} = \sum_{j=1}^n x_j y_j\)

Component-wise comparison of the vectors is defined as \({\bf x} \le {\bf y} \iff x_j \le y_j\) for all \(j\in\{1,\dots,N\}\)

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\(\delta\).2#

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 \(x_n^p \to x^p\)

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\(\delta\).3#

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.

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\(\delta\).4#

Let \(\beta \in (0,1)\). Show that the utility function \(u(c_1,c_2) = \sqrt{c_1} + \beta \sqrt{c_2}\) from \([0,\infty) \times [0,\infty)\) to \(\mathbb{R}\) to \(\mathbb{R}\) is continuous everywhere in its domain.

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\(\delta\).5#

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.

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\(\delta\).6#

Let \({\bf x}, {\bf y} \in \mathbb{R}^N\) and \(\| {\bf x} \|\) denote the Euclidean norm. Verify the Parallelogram Equality given by

\[ \| {\bf x} + {\bf y} \|^2 + \| {\bf x} - {\bf y} \|^2 = 2 \big( \| {\bf x} \|^2 + \| {\bf y} \|^2 \big) \]

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