🔬 Tutorial problems theta

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.

\(\theta\).1

Formulation

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 )

Solution

⏱

\(\theta\).2

Formulation

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)

Solution

⏱

\(\theta\).3

Formulation

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.

Solution

⏱

\(\theta\).4

Formulation

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 \(El_m D > 1\) (in which case the good is a “luxury”, whereas it is a “necessity” if \(El_m D < 1\)).

Solution

⏱

\(\theta\).5

Formulation

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?

Solution

⏱

\(\theta\).6 🍹

Formulation

Find the partial elasticities of \(z\) w.r.t. \(x\) and \(y\) in the following cases:

(a) \(z = x^3y^{-4}\)

(b) \(z = ln(x^2 + y^2)\)

(c) \(z = e^{x + y}\)

(d) \(z = (x^2 + y^2)^{1/2}\)

Solution

⏱