PoE Formula Sheet

Principles of Economics — Shortcut Sheet

This sheet focuses on “shortcuts” for common exam-style functions (Cobb-Douglas, CES, Quasilinear) to bypass long Lagrangian derivations.

1. Microeconomics Shortcuts

Functional Shortcuts (MRS & MRTS)

Function Type	Formula	MRS / MRTS Shortcut
Cobb-Douglas	$q_1^{\alpha }{q}_2^{\beta }$	$\frac{\alpha }{\beta }\cdot \frac{q_2}{q_1}$
Simplified CES	$q_1^{\alpha }+q_2^{\alpha }$	$(q_1/q_2{)}^{\alpha -1}$
Quasilinear	$f(q_1)+a{q}_2$	$f^{\prime }(q_1)/a$
Linear	$a{q}_1+b{q}_2$	$a/b$ (Constant)

Optimal Consumption Bundles ($q^{\ast }$)

Given budget constraint $y=p_1{q}_1+p_2{q}_2$

• Cobb-Douglas ($q_1^{\alpha }{q}_2^{\beta }$):
  • $q_1^{\ast }=\frac{\alpha }{\alpha +\beta }\cdot \frac{y}{p_1}$
  • $q_2^{\ast }=\frac{\beta }{\alpha +\beta }\cdot \frac{y}{p_2}$
  • Rule: Spend a fixed share $\frac{\alpha }{\alpha +\beta }$ of your income on good 1.
• Quasilinear ($f(q_1)+a{q}_2$):
  • 1. Solve $\frac{f^{\prime }(q_1)}{a}=\frac{p_1}{p_2}$ for $q_1^{\ast }$.
  • 2. $q_2^{\ast }=\frac{y-p_1{q}_1^{\ast }}{p_2}$
  • Rule: All extra income goes to good 2.

Production & Costs

• Optimal Inputs for Cobb-Douglas ($(LK{)}^{\alpha }$):
  • $L^{\ast }={\left(\frac{r}{w}\right)}^{1/2}\cdot q^{1/2\alpha }$
  • $K^{\ast }={\left(\frac{w}{r}\right)}^{1/2}\cdot q^{1/2\alpha }$
• Returns to Scale (RTS) for Cobb-Douglas:
  • $L^{\alpha }{K}^{\beta }⟹$ RTS is $\alpha +\beta$.
• Cost Function Thresholds ($C(q)=a{q}^2+bq+c^f$):
  • Min AC Price ($p_{min}$): $2\sqrt{a{c}^f}+b$
  • Min AC Quantity ($q^{\ast }$): $\sqrt{c^f/a}$
• Monopoly Revenue:
  • For linear demand $p(Q)=a-bQ$, Marginal Revenue is always $MR=a-2bQ$ (same intercept, double slope).

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2. Macroeconomics Shortcuts

Economic Growth (Solow Model)

For Cobb-Douglas Production $Y=L^{\alpha }{K}^{1-\alpha }$

• Intensive Form: $f(k)=k^{1-\alpha }$
• Steady State Capital ($k^{\ast }$): $k^{\ast }={\left(\frac{s}{\delta +n}\right)}^{1/\alpha }$
• Steady State Output ($y^{\ast }$): $y^{\ast }=(k^{\ast }{)}^{1-\alpha }$
• Golden Rule Savings Rate: $s_{gold}=1-\alpha$ (The capital share)
• Golden Rule Capital: $k_{gold}^{\ast }={\left(\frac{1-\alpha }{\delta +n}\right)}^{1/\alpha }$

Economic Fluctuations (Multipliers)

Assuming constant $MPC=dC/d(Y-T)$

• Govt. Spending Multiplier: $\frac{1}{1-MPC}$
• Tax Multiplier: $\frac{-MPC}{1-MPC}$
• Balanced Budget Multiplier: $\Delta G=\Delta T⟹\Delta Y=\Delta G$ (The multiplier is exactly 1).

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3. Quick Elasticity Rules

• Unit Elasticity: If $U=\ln (q_1)+\ln (q_2)$, price elasticity is exactly -1.
• Income Elasticity:
  • $>1$: Luxury good
  • $0\dots 1$: Necessity
  • $<0$: Inferior good