Consumption decisions: what people consume and how much.

To get to a choice, we need …

1. preference/utility: $U(x_1,x_2,\dots,x_n)$.
2. income: $m$.
3. prices: $(p_1,p_2,\dots,p_n)$

Consider $N=2$ types of goods, so the set of alternatives is $X=X_1\times X_2=[0,+\infty)\times[0,+\infty)$, and each alternative is a bundle $(x_1,x_2)$.

• Budget set: $B(m,p_1,p_2,\dots)=\{(x_1,x_2,\dots)\in X | \sum\limits_{i=1}{x_ip_i}\le m\}$
  • Convex
    • Proof:
    • In words: if you can afford bundles $\boldsymbol x$ and bundle $\boldsymbol y$, then you can afford the bundle $\alpha\boldsymbol x+(1-\alpha)\boldsymbol y$.
• Distortions: tax 分段

$$ \mathrm{max}\ U(x_1,\dots,x_n)\\

$$

• Indifference curves: $\{(x_1,x_2):U(x_1,x_2)=a\}$

• $U(x_1,x_2)=\sqrt{2\ln{x_1}+4\ln{x_2}}+100$ → $V(x_1,x_2)=x_1^2x_2^4$

• $U(x_1,x_2)=\min{\{x_1,4x_2\}}+2x_1x_2$→ $V$

indifferent ≠ no difference