1. Preferences & Rationality Weeks 1–2
A preference relation $\succsim$ encodes "at least as good as" without any numbers. From it, we derive strict preference $\succ$ and indifference $\sim$. Indifference curves are level sets of preferences.
Moving northeast is always better (monotonicity). Bowed curves toward the origin mean consumers prefer variety (convexity).
Four axioms: (1) Completeness — $x\succsim y$ or $y\succsim x$; (2) Transitivity — no cycles; (3) Continuity — upper/lower contour sets are closed; (4) Monotonicity — $x\gg y\Rightarrow x\succ y$.
WARP: If $x$ is chosen when $y$ was affordable, then whenever $y$ is chosen, $x$ must not be affordable. WARP $\Leftrightarrow$ Slutsky matrix NSD $\Leftrightarrow$ compensated demand slopes down.
Revealed preference builds up preference rankings purely from observed choices without assuming a utility function.
| Symbol | Reads as |
|---|---|
| $x\succsim y$ | $x$ at least as good as $y$ |
| $x\succ y$ | $x$ strictly preferred to $y$ |
| $x\sim y$ | $x$ indifferent to $y$ |
| $u(x)$ | Any representation of $\succsim$ |
| $\text{MRS}$ | $-\Delta y/\Delta x$ along IC |
2. Utility Functions & Indifference Curves
| Type | Form | MRS | Shape |
|---|---|---|---|
| Cobb-Douglas | $x^\alpha y^{1-\alpha}$ | $\alpha y/[(1-\alpha)x]$ | Smooth hyperbola |
| Perfect Substitutes | $ax+by$ | $a/b$ constant | Straight lines |
| Perfect Complements | $\min(ax,by)$ | undefined at kink | L-shaped |
| Quasilinear | $v(x)+y$ | $v'(x)$, no $y$ | Parallel vertical |
For Cobb-Douglas $u=x^\alpha y^{1-\alpha}$: $\text{MRS}=\alpha y/[(1-\alpha)x]$ — decreasing in $x$ (diminishing MRS, convex preferences). Corner solutions arise for perfect substitutes when $a/b \neq p_x/p_y$.
CES utility $u=[\alpha x^{(\sigma-1)/\sigma}+(1-\alpha)y^{(\sigma-1)/\sigma}]^{\sigma/(\sigma-1)}$ nests all cases: $\sigma\to 1$ = Cobb-Douglas, $\sigma\to\infty$ = perfect substitutes, $\sigma\to 0$ = Leontief. Elasticity of substitution $\sigma = -d\ln(x/y)/d\ln(\text{MRS})$.
3. Marshallian Demand — The UMP Weeks 2–3
Maximize utility on the budget line $p_x x+p_y y=m$. Solution: indifference curve tangent to budget line, i.e., $\text{MRS}=p_x/p_y$.
$\mathcal{L}=u(x,y)-\lambda(p_x x+p_y y-m)$. FOCs: $u_x=\lambda p_x$, $u_y=\lambda p_y\Rightarrow u_x/u_y=p_x/p_y$.
Roy's Identity: $x^*(p,m)=-(\partial v/\partial p_x)/(\partial v/\partial m)$. For CD: $v(p,m)=m\cdot\alpha^\alpha(1-\alpha)^{1-\alpha}/p_x^\alpha p_y^{1-\alpha}$. Apply Roy's to verify $x^*=\alpha m/p_x$.
4. Duality — Expenditure Function & Hicksian Demand Week 4
Dual question: what is the minimum cost to reach utility $\bar{u}$? Answer: the expenditure function $e(p,\bar{u})$ and Hicksian (compensated) demand $h(p,\bar{u})$. Hicksian demand moves along an IC; it isolates the pure substitution effect.
The expenditure function is: non-decreasing in $(p,\bar{u})$; homogeneous degree 1 in $p$; concave in $p$ (Hessian $\partial^2 e/\partial p_i\partial p_j$ NSD). The NSD condition is equivalent to Slutsky NSD — Hicksian demand slopes down.
Welfare: $CV=e(p^1,u^0)-m$; $EV=m-e(p^0,u^1)$. Areas under Hicksian demand curves.
5. The Slutsky Equation Week 4
Price rise → two simultaneous effects: SE (substitute toward cheaper good, always negative) and IE (real income falls, sign depends on good type). Giffen good: IE dominates, demand rises with price.
From $x^*(p,e(p,\bar{u}))=h(p,\bar{u})$, differentiate w.r.t. $p_x$:
Slutsky matrix $S_{ij}=\partial h_i/\partial p_j$: symmetric (Young's theorem on $e$), NSD, and satisfies $\sum_j S_{ij}p_j=0$ (Euler homogeneity). NSD $\Leftrightarrow$ own substitution effect $S_{ii}\leq 0$ always.
6. Elasticity & Tax Incidence Week 5
Elasticity = % response / % change. Tax incidence depends on who is less elastic. More inelastic side bears more of the tax, regardless of who legally pays it.
With tax $t$ on producers: $D(p)=S(p-t)$. Define $F(p,t)=D(p)-S(p-t)=0$. IFT gives:
IFT requires $F_p=D_p-S_p\neq 0$ (always true since $D_p<0$, $S_p>0$). Welfare: $\Delta CS+\Delta PS=-t\cdot Q^*-DWL$ where $DWL=\frac{1}{2}t^2/(|D_p|^{-1}+S_p^{-1})$ for linear curves.
7. Intertemporal Choice & The Euler Equation Week 5
Two periods, income $(y_0,y_1)$, interest rate $r$. PV budget: $c_0+c_1/(1+r)=W$. Discount factor $\beta=1/(1+\rho)$ reflects impatience.
CRRA $u=c^{1-\rho}/(1-\rho)$: consumption growth $c_1/c_0=[\beta(1+r)]^{1/\rho}$. EIS $=1/\rho$. Only permanent income shocks affect consumption (Permanent Income Hypothesis).
8. Risk & Expected Utility Week 6
Risk-averse people prefer sure $E[X]$ over the gamble $X$. Jensen's Inequality for concave $u$: $E[u(X)] \leq u(E[X])$. Risk premium = certainty equivalent gap.
Arrow-Pratt ARA: $A(w)=-u''(w)/u'(w)$. CARA: $u=-e^{-\gamma w}$, $A(w)=\gamma$.
MGF proof: $E[-e^{-\gamma\tilde{W}}]=-e^{-\gamma W_0}E[e^{-\gamma\tilde{X}}]=-e^{-\gamma W_0}e^{-\gamma\mu+\gamma^2\sigma^2/2}$. Pratt approximation exact here: $\pi=\frac{1}{2}A(w)\sigma^2=\frac{\gamma}{2}\sigma^2$.
9. Production & Cost Minimization Weeks 7–8
Minimize input costs $wL+rK$ subject to hitting output $q=f(K,L)$. Tangency: $\text{MRTS}=w/r$ (isoquant tangent to isocost). Solution gives conditional factor demands and the cost function $c(w,r,q)$.
Cost function properties: non-decreasing in $(w,r,q)$; HD1 in $(w,r)$; concave in $(w,r)$. Concavity $\Rightarrow$ factor demand slopes down.
The Hessian of $c$ w.r.t. $(w,r)$ is NSD — exactly the matrix that gives concavity. Duality: the cost function completely encodes the production technology (Shephard-Uzawa). Hotelling's Lemma: $\partial\pi^*(p)/\partial p=y^*(p)$.
10. Profit Maximization & IFT Comparative Statics Week 8
Competitive firm sets $p=MC$. SOC: diminishing MP. Long-run: free entry drives $\pi\to 0$, $p^*=\min AC$.
The IFT applies here because SOC guarantees $F_L=p\cdot f_{LL}<0\neq 0$. The denominator sign always comes from the SOC — this is the general principle. For any comparative static: sign$(\partial x^*/\partial\alpha)=$sign$(-F_\alpha)$ when SOC gives $F_x<0$.
11. Competitive Equilibrium Week 9
Market clears: $D(p^*)=S(p^*)$. Tax shifts supply; incidence depends on elasticities. Long-run: $n$ adjusts until $\pi=0$.
Walras' Law: $\sum_j p_j z_j(p)=0$ identically (value of excess demands = 0). Lets you drop one market-clearing equation. In LR competitive equilibrium, only input-price changes shift the supply curve; demand shocks change $n$ not $p^*$.
12. Monopoly & Price Discrimination Week 10
Monopolist sets $MR=MC$. Since $MR < P$, monopoly price exceeds competitive price, quantity is lower, and there is deadweight loss.
DWL$=\int_{Q^M}^{Q^C}[P(Q)-MC]dQ$. 1st-degree PD: no DWL but all surplus goes to firm. 2nd-degree: IC and IR constraints bind; high-type gets info rent; low-type distorted downward.
13. Oligopoly & Game Theory Weeks 10–11
| Model | Instrument | Result |
|---|---|---|
| Cournot | Quantities | $P>MC$, more firms → competitive |
| Stackelberg | Quantities (seq.) | Leader earns more |
| Bertrand | Prices | $P=MC$ even with 2 firms |
Stackelberg: Backward induction. Leader $q_1^*=(a-c)/(2b)$, follower $q_2^*=(a-c)/(4b)$.
NE: strategy profile where no player can profitably deviate. Nash (1950): every finite game has a NE in mixed strategies. Bertrand paradox requires homogeneous goods + equal costs. Differentiated Bertrand → $P>MC$.
14. Walrasian General Equilibrium Week 11
Edgeworth box: fixed total endowments $(\bar\omega_x,\bar\omega_y)$. Pareto improvements exist while MRS's differ. Contract curve = all PO allocations ($MRS^A=MRS^B$).
FWT: Every WE is PO. Proof: at WE, $MRS^A=p_x/p_y=MRS^B$ for all consumers → on contract curve.
SWT: Any PO allocation achievable as WE after lump-sum redistribution (requires convexity).
15. Robinson Crusoe Economy Week 12
One consumer = one firm. Planner picks how much to work to maximize $U(\ell,c)$ subject to $c=f(T-\ell)$. Planner optimum: $MRS_{\ell,c}=f'(L)$.
Firm: $pf'(L)=w\Rightarrow f'(L)=w/p$. Consumer: $U_\ell/U_c=w/p$. Both give $MRS=w/p=f'(L^*)$. Decentralized equilibrium = planner's solution = First Welfare Theorem.
With share $\theta$ of firm: consumer wealth = $wT+\theta\pi^*$. Market clearing: $c^*=f(L^*)$, $L^*=T-\ell^*$. Extension to multiple consumers proves FWT in production economies.
16. Externalities & Pigouvian Taxation Week 11
Negative externality: firm doesn't pay the full social cost → overproduces. Pigouvian tax internalizes the harm and restores the social optimum.
Coase Theorem: if property rights well-defined + zero transaction costs → bargaining achieves social optimum regardless of rights assignment. Samuelson condition for public goods: $\sum_i MB_i=MC_G$.
17. Adverse Selection — Lemons Model Week 12
Sellers know quality; buyers don't. At average-quality price, only lemons offered → price drops → more lemons → market unravels. Gains from trade in high-quality goods unrealized.
If $q < q^*$: only lemons trade — market failure despite $v_H > c_H$.
Spence signaling: separating equilibrium when $c_H(e^*)\leq\Delta w\leq c_L(e^*)$. Rothschild-Stiglitz screening: menu with IC and IR constraints; low-type gets distorted quantity; high-type earns information rent.
18. Behavioral Economics Week 12
People deviate systematically from rationality: present bias (overweight immediate), loss aversion (losses hurt ~2× more than gains), default effects (opt-out vs opt-in dramatically changes behavior).
Prospect theory: Reference-dependent, loss-averse, probability-weighted. $v(x)=x^\alpha$ (gains), $-\lambda(-x)^\alpha$ (losses), $\lambda\approx 2.25$.
Libertarian paternalism (Thaler-Sunstein): preserve choice, architect defaults to improve outcomes. Pigouvian taxes correct externalities; nudges correct internalities (self-inflicted harms from bounded rationality). System 1 vs. System 2 (Kahneman).
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Compiled reference documents for Econ 101A. All written to complement the interactive sections above.