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

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

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

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

(d) $TC=100+5X_1+6X_2-0.2X_1\sqrt{X_2}$;

(e) $U=5\ln \left(X_1\right)+2\ln \left(X_2\right)$;

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

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

(h) $q_1=100p_1^{-0.6}{p}_2^{0.4}\sqrt{Y}$;

(i) $q_2=50p_1^{-0.3}{p}_2^{-0.5}$; and

(j) $\pi =-90+20q_1^2+5q_2^2-8q_1-5q_1{q}_2$.

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

⏱

Use the chain rule to find the following partial derivatives.

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

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

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

(d) $\frac{dz}{dt}$ and $\frac{dz}{ds}$ 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)

⏱

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 $D(p,q)=a-bp{q}^{-\alpha }$, where $a,b$ and $\alpha$ are positive constants with $\alpha <1$.

Find $D_p^{\prime }(p,q)$ and $D_q^{\prime }(p,q)$, and comment on the signs of the partial derivatives.

⏱

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 $E{l}_{m}D>1$ (in which case the good is a “luxury”, whereas it is a “necessity” if $E{l}_{m}D<1$).

⏱

According to a study of industrial fishing, the annual herring catch is given by the production function $Y(K,S)=0.06157K^{1.356}{S}^{0.562}$ 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?

⏱

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

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

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

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

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

⏱