Consumer Theory

Micro

Framework

Some basic settings:

• Prices: $(p_1,\dots ,p_L)\in {\mathbb{R}}^L$
• Consumption bundles: $x=(x_1,\dots ,x_L)\in {\mathbb{R}}^L$
• Budgest $w\ge 0$
• The demand curve: $x(p_1\dots p_L,w)\in {\mathbb{R}}^L$

A simple explanation for Consumption Bundle: A consumption bundle is a specific combination of goods and services that a consumer chooses to purchase and consume. It represents the quantities of different items that make up the total consumption of an individual or household. It could be understood like there are $L$ different goods in the market, and a consumption bundle specifies how much of each good the consumer decides to buy. For example, a consumption bundle might consist of 2 units of good 1, 3 units of good 2, and 1 unit of good 3, represented as $(2,3,1)$. $x_i$ here means quantity.

Thus it is easy to come up with a Constrained Utility Maximization problem. The constraint here is your budget constraint:

$$x(p_1,\dots ,p_L)\in \underset{x\in {\mathbb{R}}_{+}^L}{\max }u(x)\text{s.t.}p\cdot x\le w$$

the constraint could also be written as ${\sum }_{\ell =1}^L{x}_{\ell }{p}_{\ell }\le w$, where $u:{\mathbb{R}}_{+}^L\to \mathbb{R}$ is the utility function.

Definition

Utility is quasi-concave if $u(\alpha x+(1-\alpha )y)\ge \min u(x),u(y)$ for all $\alpha \in [0,1]$

This definition could also be found here.

There is an exercise that to show that strict quasi-concavity implies that there exists a unique utility-maximizing demand function $x^{\ast }(p,w)$.

Proposition

Suppose that $u:{\mathbb{R}}_{+}^L\to \mathbb{R}$ is strictly quasi-concave and continuous. Then demand $x(p,w)$ is continuous for all strictly positive price vectors $(p_1,\dots ,p_L):p_{\ell }>0,\forall \ell$

Concavity and Convexity

What we are talking about while we are saying a function is whether Concave or Convexity?

Formally, we have

$$f(\alpha x+(1-\alpha )y)\ge \alpha f(x)+(1-\alpha )f(y)\text{for all }\alpha \in [0,1]$$

If a function satisfies this property, then we say it is Concave.

Similarly, if a function satisfies the opposite property:

$$f(\alpha x+(1-\alpha )y)\le \alpha f(x)+(1-\alpha )f(y)\text{for all }\alpha \in [0,1]$$

Then we say it is Convex.

From the derivatice

Quasi

We then have Quasi-Concave and Quasi-Convex, which are the weaker versions of Concave and Convex.