📌 Supply and Demand

Demand Function (Linear):

\( Q_d = a - bP \)

Supply Function (Linear):

\( Q_s = c + dP \)

Total Revenue:

\( TR = P \cdot Q \)

Average Revenue:

\( AR = \frac{TR}{Q} = P \) (perfect competition)

Marginal Revenue (Linear Demand):

\( MR = a - 2bQ \)

Total Expenditure:

\( TE = P \cdot Q \)

Consumer Surplus:

\( CS = \int_{P_e}^{\infty} Q_d(P) \, dP \) or \( CS = \frac{1}{2} (P_{\text{max}} - P_e) Q_e \) (linear)

Producer Surplus:

\( PS = \int_{-\infty}^{P_e} Q_s(P) \, dP \) or \( PS = \frac{1}{2} (P_e - P_{\text{min}}) Q_e \) (linear)

Total Surplus:

\( TS = CS + PS \)

Demand Shift (Parallel):

\( Q_d' = Q_d + k \) (where \( k \) is shift amount)

Supply Shift (Parallel):

\( Q_s' = Q_s + m \)

📌 Elasticity

Price Elasticity of Demand:

\( E_d = \left| \frac{\%\ \Delta Q_d}{\%\ \Delta P} \right| = \left| \frac{\Delta Q_d / Q_d}{\Delta P / P} \right| \)

Point Elasticity of Demand:

\( E_d = \left| \frac{dQ_d}{dP} \cdot \frac{P}{Q_d} \right| \)

Price Elasticity of Supply:

\( E_s = \frac{\%\ \Delta Q_s}{\%\ \Delta P} = \frac{\Delta Q_s / Q_s}{\Delta P / P} \)

Point Elasticity of Supply:

\( E_s = \frac{dQ_s}{dP} \cdot \frac{P}{Q_s} \)

Income Elasticity of Demand:

\( E_I = \frac{\%\ \Delta Q_d}{\%\ \Delta I} = \frac{\Delta Q_d / Q_d}{\Delta I / I} \)

Cross-Price Elasticity of Demand:

\( E_{xy} = \frac{\%\ \Delta Q_x}{\%\ \Delta P_y} = \frac{\Delta Q_x / Q_x}{\Delta P_y / P_y} \)

Elasticity and Total Revenue:

\( \Delta TR = Q (1 + E_d) \Delta P \) (if \( E_d < -1 \), elastic; \( E_d = -1 \), unitary)

Arc Elasticity of Demand:

\( E_d = \left| \frac{(Q_2 - Q_1) / \left( \frac{Q_1 + Q_2}{2} \right)}{(P_2 - P_1) / \left( \frac{P_1 + P_2}{2} \right)} \right| \)

Elasticity of Linear Demand:

\( E_d = \left| \frac{b P}{a - bP} \right| \) (for \( Q_d = a - bP \))

Elasticity at Midpoint (Linear):

\( E_d = 1 \) when \( P = \frac{a}{2b} \)

📌 Market Equilibrium

Equilibrium Condition:

\( Q_d = Q_s \)

Equilibrium Price (Linear):

\( P_e = \frac{a - c}{d + b} \) (for \( Q_d = a - bP \), \( Q_s = c + dP \))

Equilibrium Quantity (Linear):

\( Q_e = a - b \left( \frac{a - c}{d + b} \right) \)

Price Ceiling Impact:

\( Q_d > Q_s \) if \( P_{\text{ceiling}} < P_e \)

Price Floor Impact:

\( Q_s > Q_d \) if \( P_{\text{floor}} > P_e \)

Deadweight Loss (Tax):

\( DWL = \frac{1}{2} t (Q_e - Q_t) \) (where \( t \) is tax per unit)

Tax Incidence (Demand):

\( \Delta P_d = t \cdot \frac{E_s}{E_s - E_d} \)

Tax Incidence (Supply):

\( \Delta P_s = t \cdot \frac{|E_d|}{E_s - E_d} \)

Subsidy Benefit (Consumers):

\( \Delta P_d = s \cdot \frac{E_s}{E_s + |E_d|} \) (where \( s \) is subsidy)

Subsidy Benefit (Producers):

\( \Delta P_s = s \cdot \frac{|E_d|}{E_s + |E_d|} \)

Market Clearing Time:

\( t = \frac{Q_d - Q_s}{k} \) (where \( k \) is adjustment speed)

📌 Gross Domestic Product (GDP)

Expenditure Approach:

\( GDP = C + I + G + (X - M) \)

Income Approach:

\( GDP = W + R + i + P \)

Nominal GDP:

\( GDP_{\text{nominal}} = \sum (P_t \cdot Q_t) \)

Real GDP:

\( GDP_{\text{real}} = \sum (P_{\text{base}} \cdot Q_t) \)

GDP Deflator:

\( \text{Deflator} = \frac{GDP_{\text{nominal}}}{GDP_{\text{real}}} \times 100 \)

GDP Growth Rate:

\( g = \frac{GDP_{t+1} - GDP_t}{GDP_t} \times 100 \)

Per Capita GDP:

\( GDP_{\text{per capita}} = \frac{GDP}{\text{Population}} \)

Output Gap:

\( \text{Gap} = GDP_{\text{potential}} - GDP_{\text{actual}} \)

Savings Rate:

\( s = \frac{S}{GDP} \) (where \( S = GDP - C - G \))

Multiplier Effect:

\( k = \frac{1}{1 - MPC} \) (where \( MPC \) is marginal propensity to consume)

Investment Multiplier:

\( \Delta GDP = k \cdot \Delta I \)

📌 Inflation

Inflation Rate:

\( \pi = \frac{CPI_{t+1} - CPI_t}{CPI_t} \times 100 \)

Consumer Price Index (CPI):

\( CPI = \frac{\sum (P_t \cdot Q_{\text{base}})}{\sum (P_{\text{base}} \cdot Q_{\text{base}})} \times 100 \)

Real Value Adjustment:

\( \text{Real Value} = \frac{\text{Nominal Value}}{1 + \pi} \)

Purchasing Power:

\( PP = \frac{1}{CPI} \times \text{Base Year Value} \)

Fisher Effect:

\( i = r + \pi^e \) (where \( i \) is nominal rate, \( r \) is real rate, \( \pi^e \) is expected inflation)

Inflation-Adjusted Return:

\( r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} - 1 \)

Quantity Theory of Money:

\( MV = PY \) (where \( M \) is money supply, \( V \) is velocity, \( P \) is price level, \( Y \) is real GDP)

Inflation from Money Growth:

\( \pi = g_M + g_V - g_Y \) (growth rates of \( M \), \( V \), \( Y \))

Cost-Push Inflation:

\( \pi = \frac{\Delta TC}{Q} \) (increase in total cost per unit output)

Demand-Pull Inflation:

\( \pi = f(\Delta AD) \) (function of aggregate demand increase)

📌 Cost Analysis

Total Cost:

\( TC = FC + VC \)

Average Total Cost:

\( ATC = \frac{TC}{Q} \)

Average Fixed Cost:

\( AFC = \frac{FC}{Q} \)

Average Variable Cost:

\( AVC = \frac{VC}{Q} \)

Marginal Cost:

\( MC = \frac{\Delta TC}{\Delta Q} \)

Profit Maximization (Perfect Competition):

\( MC = MR = P \)

Profit:

\( \pi = TR - TC \)

Break-Even Quantity:

\( Q_{\text{break-even}} = \frac{FC}{P - AVC} \)

Cost Function (Linear):

\( TC = F + vQ \) (where \( F \) is fixed, \( v \) is variable cost per unit)

Economies of Scale:

\( ATC \) decreases as \( Q \) increases

Long-Run Average Cost:

\( LRAC = \frac{LRTC}{Q} \) (long-run total cost)

📌 Utility and Consumer Behavior

Utility Function:

\( U = U(x, y) \) (e.g., \( U = x^\alpha y^\beta \))

Marginal Utility:

\( MU_x = \frac{\partial U}{\partial x} \)

Law of Diminishing Marginal Utility:

\( \frac{\partial MU_x}{\partial x} < 0 \)

Budget Constraint:

\( P_x x + P_y y = I \)

Utility Maximization:

\( \frac{MU_x}{P_x} = \frac{MU_y}{P_y} \)

Demand Curve from Utility:

\( x = \frac{I}{P_x} \cdot \frac{MU_x}{MU_x + MU_y} \) (Cobb-Douglas example)

Indifference Curve Slope (MRS):

\( MRS = -\frac{MU_x}{MU_y} \)

Consumer Equilibrium:

\( MRS = \frac{P_x}{P_y} \)

Income Effect:

\( \Delta x_I = x(I + \Delta I) - x(I) \)

Substitution Effect:

\( \Delta x_S = x(P_x', I_{\text{comp}}) - x(P_x, I) \)

Total Effect:

\( \Delta x = \Delta x_S + \Delta x_I \)

📌 Game Theory

Payoff (2-Player Game):

\( \pi_i = f(s_i, s_{-i}) \) (where \( s_i \) is strategy of player \( i \))

Nash Equilibrium Condition:

\( \pi_i(s_i^*, s_{-i}^*) \geq \pi_i(s_i, s_{-i}^*) \) for all \( s_i \)

Expected Payoff (Mixed Strategy):

\( E(\pi_i) = \sum p_j \pi_i(s_i, s_j) \) (where \( p_j \) is probability of strategy \( s_j \))

Minimax Value:

\( v_i = \min_{s_{-i}} \max_{s_i} \pi_i(s_i, s_{-i}) \)

Dominant Strategy Payoff:

\( \pi_i(s_i^*, s_{-i}) \geq \pi_i(s_i, s_{-i}) \) for all \( s_{-i} \)

Zero-Sum Game:

\( \pi_1 + \pi_2 = 0 \)

Best Response Function:

\( BR_i(s_{-i}) = \arg\max_{s_i} \pi_i(s_i, s_{-i}) \)

Cournot Duopoly Output:

\( q_i = \frac{a - c - b q_{-i}}{2b} \) (linear demand \( P = a - bQ \))

Bertrand Price Competition:

\( P_i = c \) (marginal cost in perfect competition)

Stackelberg Output (Leader):

\( q_1 = \frac{a - c}{2b} \) (linear demand)

📌 Econometrics

Simple Linear Regression:

\( Y = \beta_0 + \beta_1 X + \epsilon \)

Slope (\( \beta_1 \)):

\( \beta_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \)

Intercept (\( \beta_0 \)):

\( \beta_0 = \bar{Y} - \beta_1 \bar{X} \)

Coefficient of Determination (\( R^2 \)):

\( R^2 = 1 - \frac{\sum (Y_i - \hat{Y}_i)^2}{\sum (Y_i - \bar{Y})^2} \)

Standard Error of Estimate:

\( SE = \sqrt{\frac{\sum (Y_i - \hat{Y}_i)^2}{n - 2}} \)

t-Statistic:

\( t = \frac{\beta_1}{SE(\beta_1)} \)

F-Statistic (ANOVA):

\( F = \frac{\text{MSR}}{\text{MSE}} \) (regression vs. error mean squares)

Autocorrelation (Durbin-Watson):

\( DW = \frac{\sum (e_t - e_{t-1})^2}{\sum e_t^2} \) (where \( e_t \) is residual)

Log-Linear Model:

\( \ln Y = \beta_0 + \beta_1 \ln X + \epsilon \) (elasticity interpretation)

Time Series Growth Rate:

\( g = \frac{Y_{t+1} - Y_t}{Y_t} \) or \( g = e^{\beta_1} - 1 \) (log model)