We will call the possible goods and services that a consumer can consume as commodities. A consumer can consume finite possible commodities and each of those commodities can be defined as ${{x}_{1}},{{x}_{2}}\cdots ,{{x}_{L}}$. This set can be defined as the commodity vector or commodity bundle can be viewed as the point in ${{\mathbb{R}}^{L}}$ or the real commodity space and given as: \[x=\left[ \begin{matrix} {{x}_{1}} \\ \vdots \\ {{x}_{L}} \\ \end{matrix} \right]\] The consumption set $X$ is the subset of such ${L}$ dimensional real commodity space ${{\mathbb{R}}^{L}}$ i.e. $X\subset {{\mathbb{R}}^{L}}$. The $X$ is subset or limited because of various physical constraints imposed by the environment. The consumption set is defined as: \[X={{\mathbb{R}}^{L}}=\left\{ x\in {{\mathbb{R}}^{L}}:{{x}_{L}}\ge 0\forall l=1,2,\cdots L \right\}\] Thus it has major two properties: a) this set is non-negative bundles of commodities and b) this set is convex. One major constraints imposed in the consumption set is affordability. Such affordability depends upon two major things. a) prices of those commodities $p=\left( {{p}_{1}},\cdots {{p}_{L}} \right)$ and b) consumer wealth $w$. Each of these commodities follows principle of completeness or universality, formally, each of these commodities are traded in market at prices quoted publicly and each prices are strictly positive i.e $p>>0$. Further, $p\centerdot x={{p}_{1}}{{x}_{1}}+\cdots +{{p}_{L}}{{x}_{L}}=\sum\limits_{l=1}^{L}{{{p}_{l}}{{x}_{l}}}\le w$ implies the feasibility of such consumption set. Then we can define, Walrasian or competitive budget set as ${{B}_{p,w}}=\left\{ x\subset \mathbb{R}_{+}^{L}:p\centerdot x\le w \right\}$. If, ${{B}_{p,w}}=\left\{ x\subset \mathbb{R}_{+}^{L}:p\centerdot x=w \right\}$then its known as the budget hyperplane. This competitive Walrasian budget set is also convex. The consumer's Walrasian (or market or ordinary) demand correspondence $x(p,w)$ assigns a set of chosen consumption bundle for each price-wealth pair and for single valued $x(p,w)$is referred to demand function. Such demand function has 3 major properties. Firstly, Walrasian demand correspondence $x(p,w)$satisfies the Walrus law i.e. $x(p,w)=w$for $p>>0\And w>0$, that means that there is no slackness or the consumer fully spend their wealth or, $\sum\limits_{l=1}^{L}{{{p}_{l}}{{x}_{l}}}=w$. For now we relax the inter-temporal concept that consumer may safe for future consumption. Secondly, the Walrasian demand correspondence is homogeneous of degree zero i.e any proportionally increment in prices and wealth will un-affect the demand or in other word only the real opportunities matters, formally, $x(\alpha p,\alpha w)=x(p,w)$for any $p,w,\alpha >0$. Lastly, the consumer reveal the information about the stable preferences which is known as the Weak Axiom of Revealed Preferences. Say when two different bundles $x$ and $y$ were available to consumer when his budget was ${{B}_{p,w}}$ and he chooses $x$(he revealed his preferences here) then in another budget say, ${{B}_{{p}',{w}'}}$, if he chooses the bundle $y$, then we can infer that $x$ was in-feasible in budget ${{B}_{{p}',{w}'}}$. For the fixed prices $\overline{p}$, the function of wealth $x(\overline{p},w)$ is known as the consumer's Engel function and its image in positive real space $\mathbb{R}_{+}^{L}$ is known as the wealth expansion path. The derivative, $\frac{\partial {{x}_{L}}(\overline{p},w)}{\partial w}$ is known as the wealth effect. Such wealth effect is positive for normal goods and negative for the inferior goods and zero for the necessity goods. The wealth effects of all the commodities of the consumption bundle real space is given in matrix notation as: \[{{D}_{w}}x(p,w)={{\left[ \begin{matrix} \frac{\partial {{x}_{1}}(p,w)}{\partial w} \\ \frac{\partial {{x}_{2}}(p,w)}{\partial w} \\ \vdots \\ \frac{\partial {{x}_{L}}(p,w)}{\partial w} \\ \end{matrix} \right]}_{L\times 1}}\in {{\mathbb{R}}^{L}}\] For the fixed wealth $\overline{w}$, the function of price $x(p,\overline{w})$ image in positive real space $\mathbb{R}_{+}^{L}$ is known as the offer curve. The derivative, $\frac{\partial {{x}_{l}}(\overline{p},w)}{\partial {{p}_{k}}}$ is known as the price effect. The own price effect is always negative (as per law of demand) i.e $\frac{\partial {{x}_{l}}(\overline{p},w)}{\partial {{p}_{k}}}<0$ $\forall$ $l=k$ except for the giffen goods which is positive i.e $\frac{\partial {{x}_{l}}(\overline{p},w)}{\partial {{p}_{k}}}<0$ for $l=k$. While, this derivative $\frac{\partial {{x}_{l}}(\overline{p},w)}{\partial {{p}_{k}}}<0$ for$l\ne k$ is positive/negative/zero iff $l$ and $k$ are substitute/complementary/unrelated for each other. The price effect in matrix notation is given in $L\times L$ dimension as: \[{{D}_{p}}x(p,w)={{\left[ \begin{matrix} \frac{\partial {{x}_{1}}(p,w)}{\partial {{p}_{1}}} & \cdots & \frac{\partial {{x}_{1}}(p,w)}{\partial {{p}_{L}}} \\ \vdots & \ddots & \vdots \\ \frac{\partial {{x}_{L}}(p,w)}{\partial {{p}_{1}}} & \cdots & \frac{\partial {{x}_{L}}(p,w)}{\partial {{p}_{L}}} \\ \end{matrix} \right]}_{L\times L}}\] where, diagonal elements are own-price effects. One major problems with the marginal are they are not unit-free therefore for different units its create the problem to compare two different marginal. Analysis of elasticity's solve such problem as the elasticity's are unit-free measure of marginal. The percentage change in quantity i.e ${\scriptstyle{}^{\partial {{x}_{l}}(p,w)}/{}_{{{x}_{l}}(p,w)}}$ w.r.t to percentage change in price ${\scriptstyle{}^{\partial {{p}_{k}}}/{}_{{{p}_{k}}}}$ gives the price elasticity of demand expressed as: ${{\varepsilon }_{lk}}(p,w)=\left( \frac{{\scriptstyle{}^{\partial {{x}_{l}}(p,w)}/{}_{{{x}_{l}}(p,w)}}}{{\scriptstyle{}^{\partial {{p}_{k}}}/{}_{{{p}_{k}}}}} \right)$ which can be re-expressed as: ${{\varepsilon }_{lk}}(p,w)=\left( \frac{\partial {{x}_{l}}(p,w)}{\partial {{p}_{k}}} \right)\left( \frac{{{p}_{k}}}{{{x}_{l}}(p,w)} \right)$ where, $\frac{\partial {{x}_{l}}(p,w)}{\partial {{p}_{k}}}$ is the slope of demand function. For, gives the own-price elasticity and for $l\ne k$gives the cross-elasticity. Similarly, the percentage change in quantity i.e ${\scriptstyle{}^{\partial {{x}_{l}}(p,w)}/{}_{{{x}_{l}}(p,w)}}$ w.r.t to percentage change in wealth ${\scriptstyle{}^{\partial w}/{}_{w}}$ gives the wealth elasticity of demand expressed as: ${{\varepsilon }_{lw}}(p,w)=\left( \frac{{\scriptstyle{}^{\partial {{x}_{l}}(p,w)}/{}_{{{x}_{l}}(p,w)}}}{{\scriptstyle{}^{\partial w}/{}_{w}}} \right)$ which can be re-expressed as: ${{\varepsilon }_{lk}}(p,w)=\left( \frac{\partial {{x}_{l}}(p,w)}{\partial w} \right)\left( \frac{w}{{{x}_{l}}(p,w)} \right)$ where, $\frac{\partial {{x}_{l}}(p,w)}{\partial w}$ $\frac{\partial {{x}_{l}}(p,w)}{\partial {{p}_{k}}}$ is the slope of income or wealth demand function.

Note: Once, I define the demand correspondence. I will write $p\centerdot x={{p}_{1}}{{x}_{1}}+\cdots +{{p}_{L}}{{x}_{L}}=\sum\limits_{l=1}^{L}{{{p}_{l}}{{x}_{l}}}\le w$ as $p\centerdot x(p,w)={{p}_{1}}{{x}_{1}}(p,w)+\cdots +{{p}_{L}}{{x}_{L}}(p,w)=\sum\limits_{l=1}^{L}{{{p}_{l}}{{x}_{l}}}(p,w)\le w$.

(Because, I saw Batman last night!)