🔬 Tutorial problems iota

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 are for additional practice. The symbol 🍹 indicates additional problems.

\(\iota\).1

Formulation

Cobb-Douglas Preferences Specific Example:

Consider the utility function \(U(x, y)=x^{0.5} y^{0.5}\) that is defined on the consumption set \(\mathbb{R}_{+}^{2}\).

(a) Find the equation of the indifference curve that corresponds to \(U=40\).

(b) What is the slope of the indifference curve for \(U=40\) for any given value of \(x\)?

(c) What is the equation of an arbitrary indifference curve for this utility function?

(d) What is the slope of an arbitrary indifference curve for this utility function at any given value of \(x\)?

Solution

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\(\iota\).2

Formulation

Marginal Rates of Substitution:

Calculate the marginal rate of substitution for an arbitrary commodity bundle of the form \((x, y)>>\) \((0,0)\) (that is, where \(x>0\) and \(y>0\) ) for each of the following utility functions.

(a) Quasi-Linear Preferences Example 1: \(U(x, y)=x+\sqrt{y}\).

(b) Quasi-Linear Preferences Example 2: \(U(x, y)=x+\ln (y)\).

(c) Stone-Geary Preferences: \(U(x, y)=\left(x-x_{0}\right)^{\alpha}\left(y-y_{0}\right)^{1-\alpha}\), where \(x_{0}>0, y_{0}>0\), and \(\alpha \in(0,1)\) are fixed parameters.

(d) Constant-Elasticity-of-Substitution (CES) Preferences: \(U(x, y)=\left(\alpha x^{\rho}+\beta y^{\rho}\right)^{\frac{1}{\rho}}\), where \(x_{0}>0, y_{0}>0\), and \(\alpha \in(0,1)\) are fixed parameters.

Solution

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\(\iota\).3

Formulation

Let \(U\left(q_{1}, q_{2}\right)\) be a utility function that represents a particular individual’s preferences over bundles of strictly positive amounts of each of two commodities. Suppose that this utility function is at least twice continuously differentiable. The indifference curve corresponding to utility level \(\hat{U}\) for this individual is a graphical representation of the level-set for this utility function that corresponds to utility-level \(\hat{U}\). It is formally defined to be

\[ U^{0}\left(q_{1}, q_{2}, \hat{U}\right)=\left\{\left(q_{1}, q_{2} \in \mathbb{R}_{++}^{2}: U\left(q_{1}, q_{2}\right)=\hat{U}\right\}\right. \]

The equation that describes this curve takes the form \(q_{2}=f\left(q_{1}\right)\), for some function \(f\). The function \(f\) is implicitly defined by the equation

\[ U\left(q_{1}, q_{2}\right)=\hat{U} \]

Use the equation \(U\left(q_{1}, q_{2}\right)=\hat{U}\) to show that the slope of the above indifference curve is given by

\[ \left.\frac{d q_{2}}{d q_{1}}\right|_{U=\hat{U}}=-M R S_{1,2}\left(q_{1}, q_{2}\right) \]

where MRS stands for the marginal rate of substitution, and is defined to be

\[ M R S_{1,2}\left(q_{1}, q_{2}\right)=\frac{\left(\frac{\partial U\left(q_{1}, q_{2}\right)}{\partial q_{1}}\right)}{\left(\frac{\partial U\left(q_{1}, q_{2}\right)}{\partial q_{2}}\right)} \]

Solution

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\(\iota\).4

Formulation

Find the degree of homogeneity, if there is one, for each of the following functions:

(a) \(f(x,y,z) = 3x+4y-3z\)

(b) \(g(x,y,z) = 3x+4y-2z-2\)

(c) \(h(x,y,z) = \frac{\sqrt{x}+\sqrt{y}+\sqrt{z}}{x+y+z}\)

(d) \(G(x,y) = \sqrt{xy} \ln \left( \frac{x^2+y^2}{xy} \right)\)

(e) \(H(x,y) = \ln x + \ln y\)

(f) \(p(x_1,\dots,x_n) = \sum_{i=1}^{n} x_i^n\)

This question comes from Sydsæter, Hammond, Strøm, and Carvajal (2016) (Section 12.7, Problem 1)

Solution

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