🔬 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#
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 )
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
\(\theta\).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)
(a)
(b)
(c)
Let \(a(r)=r, b(r)=1-r\), and \(c(r)=\frac{1}{(1-r)}\). We will assume throughout that \(r \in(-\infty, 1) \cup(1, \infty)\). Since \(g(r)=F(a(r), b(r), c(r))\), we have
(d)
\(\theta\).3#
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.
The partial derivatives are as follows:
\(D'_p(p, q) < 0\) implies that as the price \(p\) increases, the demand \(D\) will decrease, ceteris paribus.
\(D'_q(p, q) > 0\) implies that as the competing producer’s price \(q\) increases, the demand \(D\) will increase, ceteris paribus.
\(\theta\).4#
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 \(El_m D < 1\)).
Since the good is “luxury”, we have
We wish to show that \(\frac{pD}{m}\) increases with income when the above holds; that is, to show that \(\frac{\partial}{\partial m} \left(\frac{pD}{m} \right) > 0\).
We know that \(\frac{m}{D} \frac{\partial D}{\partial m} > 1 \iff \left( \frac{m}{D} \frac{\partial D}{\partial m} - 1 \right) > 0\). We further assume that \(p, m > 0\) as these are price and income respectively, which typically don’t make sense with negative or zero values. Finally, we also assume that quantity demanded \(D > 0\), noting that if it were zero, it would not make sense to have it on the denominator of the partial elasticity calculation.
Thus we have
meaning that
which is what we wanted to show.
\(\theta\).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?
(a)
(b)
In other words, the size of the catch increases by a factor of \((2)^{1.356 + 0.562} \approx 3.779\) when both inputs are scaled by a factor of two.
\(\theta\).6 🍹#
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) \(z = (x^2 + y^2)^{1/2}\)
(a)
(b)
(c)
(d)