🔬 Tutorial problems kappa \kappa

🔬 Tutorial problems kappa \(\kappa\)#

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.

\(\kappa\).1#

Find all of the first-order and second-order partial derivatives for each of the following functions.

(a) \(U=10 X^{0.6} L^{0.5}\);

(b) \(C=50 Y^{0.8} i^{0.3}\) (where \(i\) is a variable, not the imaginary number \(\sqrt{-1})\);

(c) \(C=10+2 Y^{0.9}+5 \sqrt{i}\) (where \(i\) is a variable, not the imaginary number \(\sqrt{-1}\));

(d) \(T C=100+5 X_{1}+6 X_{2}-0.2 X_{1} \sqrt{X_{2}}\);

(e) \(U=5 \ln \left(X_{1}\right)+2 \ln \left(X_{2}\right)\);

(f) \(Q=50 K^{0.7} L^{0.3}\);

(g) \(Q=A K^{\alpha} L^{1-\alpha}\);

(h) \(q_{1}=100 p_{1}^{-0.6} p_{2}^{0.4} \sqrt{Y}\);

(i) \(q_{2}=50 p_{1}^{-0.3} p_{2}^{-0.5}\); and

(j) \(\pi=-90+20 q_{1}^{2}+5 q_{2}^{2}-8 q_{1}-5 q_{1} q_{2}\).

This question comes from Shannon (1995, p. 501-502, questions 1 and 2 )

\(\kappa\).2#

Use the chain rule to find the following partial derivatives.

(a) \(\frac{d z}{d t}\) if \(z=F(x, y)=x^{2}+e^{y}\), where \(x=t^{3}\) and \(y=2 t\).

(b) \(\frac{d Y}{d t}\) if \(Y=F(L, K)=K L^{2}\), where \(L=f(t)\) and \(K=g(t)\).

(c) \(g^{\prime}(r)\) if \(g(r)=F\left(r, 1-r, \frac{1}{(1-r)}\right)\).

(d) \(\frac{d z}{d t}\) and \(\frac{d z}{d s}\) if \(z=F(x, y)\), where \(x=f(t)\) and \(y=g(t, s)\).

This question comes from the instructors manual for Sydsaeter et al (2016)

\(\kappa\).3#

Let \(D(p, m)\) indicate a typical consumer’s demand for a particular commodity, as a function of its price \(p\) and the consumer’s own income \(m\). Show that the proportion \(pD/m\) of income spent on the commodity increases with income if income elasticity of demand \(El_m D > 1\) (in which case the good is a “luxury”, whereas it is a “necessity” if \(El_m D < 1\)).

\(\kappa\).4#

In which direction should one move from a given point in order to increase the value of the function most rapidly:

  1. \(\quad\) \(f(x,y) = 4x^2y\) from the point \((2,3)\)

  2. \(\quad\) \(f(x,y) = y^2 e^{3x}\) from the point \((0,3)\)

Present your answer as a vector of length 1.

[Simon and Blume, 1994]: Exercises 14.18, 14.19

Review the definition and facts about the gradient of a multivariate functions.

\(\kappa\).5#

Consider a function \(f : \mathbb{R}^N \ni x \mapsto x'Ax \in \mathbb{R}\), where \(N \times N\) matrix \(A\) is symmetric.

Using the product rule of multivariate calculus, derive the gradient and Hessian of \(f\). Make sure that all multiplied vectors and matrices are conformable.

You can assume that \(x\) is a column vector, and that any vector function of \(x\) is also a column vector.

\(\kappa\).6#

Compute directional derivative of \(f(x,y) = xy^2 + x^3y\) at the point \((4,-2)\) in the direction of the vector \((1/\sqrt{10},3/\sqrt{10})\).

Proceed in two different ways:

  • first, using the definition of the directional derivative, write down a function \(g \colon \mathbb{R} \to \mathbb{R}\) of \(h\) given as a slice of the original function \(f(x,y)\) through the given point in the given direction; then differentiate this function and compute the derivative at \(h=0\)

  • second, use the gradient formula; and verify that the same answer is obtained

[Simon and Blume, 1994]: Exercise 14.20

Follow the example in the lecture notes.

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