🔬 Tutorial problems iota \(\iota\)#
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 out are for additional practice on your own.
\(\iota\).1#
The demand for a product, \(D\), depends on the price \(p\) of the product and on the price \(q\) charged by a competing producer. It is \(D(p, q) = a - bpq^{-\alpha}\), where \(a, b\) and \(\alpha\) are positive constants with \(\alpha < 1\).
Find \(D'_p(p, q)\) and \(D'_q(p, q)\), and comment on the signs of the partial derivatives.
\(\iota\).2#
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 \(x\)?
\(\iota\).3#
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.
\(\iota\).4#
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)
\(\iota\).5#
According to a study of industrial fishing, the annual herring catch is given by the production function \(Y(K, S) = 0.06157 K^{1.356} S^{0.562}\) involving the catching effort \(K\) and the herring stock \(S\).
(a) Find \(\frac{\partial Y}{\partial K}\) and \(\frac{\partial Y}{\partial S}\).
(b) If K and S are both doubled, what happens to the catch?