🎯 Curated Multi-Concept Problem Sets

Welcome to the Flügel Economics Analytical Problem Laboratory. High-stakes competitive examinations like the International Economics Olympiad (IEO), AP Micro/Macro, and university economics honors programs require synthesizing calculus optimization, equilibrium models, and econometric data analysis.

Try solving these problems before expanding the progressive hint scaffolds.

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⚡ Problem 1 (Microeconomics): Cournot Duopoly with Asymmetric Marginal Costs

Problem Statement

Two firms compete as Cournot duopolists producing a homogeneous product. The inverse market demand is given by:

$$P(Q)=120-Q\text{where }Q=q_1+q_2$$

• Firm 1 has a low marginal cost: $C_1(q_1)=20q_1$ ($M{C}_1=20$).
• Firm 2 has a high marginal cost: $C_2(q_2)=40q_2$ ($M{C}_2=40$). Both firms choose quantities simultaneously.

1. Derive the best-response (reaction) function for each firm: $R_1(q_2)$ and $R_2(q_1)$.
2. Calculate the Cournot-Nash equilibrium quantities $q_1^{\ast },q_2^{\ast }$, aggregate output $Q^{\ast }$, and market price $P^{\ast }$.
3. Calculate the individual profits ${\pi }_1^{\ast }$ and ${\pi }_2^{\ast }$.
4. Determine the maximum marginal cost $c_2^{max}$ for Firm 2 before it is driven out of the market ($q_2^{\ast }=0$).

💡 Interactive Clue SystemStep-by-Step Problem Solving Clues

Try solving with Hint 1 before revealing Hint 2 or 3!

🟡 Hint 1: Physical Intuition & Coordinate Choice

Set up individual profit equations $\pi_i = (120 - q_1 - q_2 - c_i) q_i$ and take partial derivatives with respect to $q_i$ to set $MR_i = MC_i$.

🟠 Hint 2: Key Governing Equations

Firm 1's reaction function is $q_1 = \frac{100 - q_2}{2}$. Firm 2's reaction function is $q_2 = \frac{80 - q_1}{2}$. Solve this $2 \times 2$ linear system simultaneously.

🟢 Hint 3: Full Step-by-Step Derivation

Substitute I_C = I_cm + M*R^2 and solve for acceleration a_cm.

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🏛️ Problem 2 (Macroeconomics): IS-LM Multiplier with Liquidity Preference

Problem Statement

Consider an economy with:

• $C=150+0.8(Y-T)$, $I=250-1000i$, $G=200$, $T=150$.
• Real money supply $\frac{M}{P}=600$, and real money demand $L(Y,i)=0.25Y-2000i$.

1. Derive the IS and LM equations.
2. Find the equilibrium levels of output $Y^{\ast }$ and interest rate $i^{\ast }$.
3. If government purchases increase by $\mathrm{\Delta }G=50$, calculate the new equilibrium output $Y^{\prime }$, new interest rate $i^{\prime }$, and the value of investment crowded out.

💡 Interactive Clue SystemStep-by-Step Problem Solving Clues

Try solving with Hint 1 before revealing Hint 2 or 3!

🟡 Hint 1: Physical Intuition & Coordinate Choice

For IS: $Y = C + I + G$. For LM: $M/P = L(Y, i)$.

🟠 Hint 2: Key Governing Equations

The initial IS curve is $Y = 2400 - 5000 i$. The LM curve is $Y = 2400 + 8000 i$.

🟢 Hint 3: Full Step-by-Step Derivation

Substitute I_C = I_cm + M*R^2 and solve for acceleration a_cm.

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🌐 Problem 3 (Quantitative): Econometric OLS Regression & $t$-Test

Problem Statement

An econometrician runs an OLS regression of CEO compensation ($\ln (\text{Salary})$) on company sales ($\ln (\text{Sales})$) and return on equity ($ROE$) with sample size $N=200$:

$$\widehat{\ln (\text{Salary})}=4.50+0.280\ln (\text{Sales})+0.015ROE,R^2=0.45$$

Standard errors are: $SE({\hat{\beta }}_{sales})=0.035$ and $SE({\hat{\beta }}_{roe})=0.005$.

1. Interpret the economic meaning of the coefficient ${\hat{\beta }}_{sales}=0.280$.
2. Test the null hypothesis $H_0:{\beta }_{sales}=0$ against $H_1:{\beta }_{sales}>0$ at the $1\mathrm{\%}$ significance level (critical $t_{0.01}=2.33$).
3. Construct a $95\mathrm{\%}$ confidence interval for the sales elasticity parameter ${\beta }_{sales}$ (critical $t_{0.025}=1.96$).

💡 Interactive Clue SystemStep-by-Step Problem Solving Clues

Try solving with Hint 1 before revealing Hint 2 or 3!

🟡 Hint 1: Physical Intuition & Coordinate Choice

In a log-log specification ($\ln Y$ on $\ln X$), the slope coefficient is an elasticity ($\% \Delta Y / \% \Delta X$).

🟠 Hint 2: Key Governing Equations

The $t$-statistic is $t = \frac{\hat{\beta} - 0}{SE(\hat{\beta})}$. The confidence interval is $\hat{\beta} \pm t_{crit} \times SE(\hat{\beta})$.

🟢 Hint 3: Full Step-by-Step Derivation

Substitute I_C = I_cm + M*R^2 and solve for acceleration a_cm.