Fundamental Economic Equations

Why This Matters

Mathematical economics isn't about memorizing formulas—it's about understanding the logical structure that connects individual decisions to market outcomes. Every equation you'll encounter represents a fundamental truth about how consumers choose, how firms produce, and how markets reach balance. You're being tested on your ability to derive, interpret, and apply these relationships, not just plug in numbers.

The equations in this guide fall into interconnected categories: equilibrium conditions, optimization rules, behavioral functions, and sensitivity measures. Master the underlying logic of each, and you'll recognize them instantly whether they appear in a multiple-choice question or an FRQ asking you to "show mathematically" why a firm should change its output. Don't just memorize the formulas—know what economic principle each equation captures and when to deploy it.

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Equilibrium and Market Clearing

Markets function because prices adjust until buyers and sellers agree. These equations define the conditions under which that balance occurs.

Supply and Demand Equilibrium

• $Q_d = Q_s$—the fundamental market-clearing condition where quantity demanded equals quantity supplied at the equilibrium price $P^*$
• Equilibrium price emerges endogenously from the interaction of supply and demand functions; solve by setting $D(P) = S(P)$ and solving for $P$
• Comparative statics allow you to predict how shifts in either curve (from taxes, preferences, or input costs) change $P^*$ and $Q^*$

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Production and Cost Theory

These equations describe how firms transform inputs into outputs and the cost structures that constrain their decisions. The key insight: production functions define what's technically possible, while cost functions define what's economically efficient.

Cobb-Douglas Production Function

• $Q = A L^\alpha K^\beta$—relates output $Q$ to labor $L$ and capital $K$, with $A$ representing total factor productivity
• Returns to scale depend on $\alpha + \beta$: constant if equal to 1, increasing if greater than 1, decreasing if less than 1
• Marginal products are $MP_L = \alpha \frac{Q}{L}$ and $MP_K = \beta \frac{Q}{K}$—essential for input optimization problems

Cost Minimization

• $\frac{MP_L}{w} = \frac{MP_K}{r}$—the tangency condition stating that the last dollar spent on each input must yield equal marginal product
• Isocost line $C = wL + rK$ represents all input combinations with the same total cost; optimal production occurs where it's tangent to the isoquant
• Expansion path traces optimal input combinations as output changes—critical for deriving long-run cost curves

Marginal Revenue and Marginal Cost

• Marginal cost $MC = \frac{dTC}{dQ}$—the additional cost of producing one more unit; typically U-shaped due to diminishing returns
• Marginal revenue $MR = \frac{dTR}{dQ}$—for price-takers, $MR = P$; for price-setters, $MR = P\left(1 + \frac{1}{\varepsilon_d}\right)$
• The MC curve above average variable cost is the firm's supply curve in perfect competition

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Consumer Choice and Optimization

Consumer theory rests on the assumption that individuals maximize utility subject to budget constraints. These equations formalize rational choice.

Utility Maximization

• $\max U(x, y)$ subject to $P_x x + P_y y = M$—the canonical consumer problem where $M$ is income and $P_x$, $P_y$ are prices
• Optimality condition: $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$, meaning the marginal utility per dollar must be equal across all goods
• Lagrangian method yields $\mathcal{L} = U(x,y) + \lambda(M - P_x x - P_y y)$—the multiplier $\lambda$ represents the marginal utility of income

Marginal Rate of Substitution

• $MRS_{xy} = \frac{MU_x}{MU_y}$—the rate at which a consumer willingly trades good $y$ for good $x$ while staying on the same indifference curve
• Geometric interpretation: the negative slope of the indifference curve at any point; diminishing MRS reflects convex preferences
• At optimum: $MRS_{xy} = \frac{P_x}{P_y}$—the subjective trade-off equals the market trade-off

Income and Substitution Effects

• Substitution effect—the change in quantity demanded due solely to the relative price change, holding utility constant (always negative for price increases)
• Income effect—the change in quantity demanded due to the change in purchasing power; direction depends on whether the good is normal or inferior
• Slutsky equation: $\frac{\partial x}{\partial P_x} = \frac{\partial x^h}{\partial P_x} - x \frac{\partial x}{\partial M}$—decomposes total effect into substitution and income components

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Firm Optimization

Profit-seeking behavior drives firm decisions. These conditions tell you exactly when a firm should expand, contract, or shut down.

Profit Maximization

• $\pi = TR - TC$ where $TR = P \cdot Q$ and $TC = FC + VC(Q)$—profit is the residual after all costs
• First-order condition: $MR = MC$—produce where the revenue from the last unit exactly covers its cost
• Second-order condition: $\frac{dMC}{dQ} > \frac{dMR}{dQ}$—ensures you're at a maximum, not a minimum; MC must cut MR from below

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Sensitivity and Responsiveness

Elasticity measures how much one variable responds to changes in another. These ratios are unit-free, making them ideal for comparisons across markets.

Elasticity of Demand

• Price elasticity: $\varepsilon_d = \frac{\% \Delta Q_d}{\% \Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$—measures responsiveness of quantity to price changes
• Elastic ($|\varepsilon_d| > 1$): price cuts raise revenue; Inelastic ($|\varepsilon_d| < 1$): price cuts lower revenue
• Total revenue test: $TR$ is maximized where $|\varepsilon_d| = 1$—the point of unit elasticity

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Aggregate Behavior

Macroeconomic equations aggregate individual decisions to describe economy-wide patterns.

Consumption Function

• $C = \bar{C} + cY_d$—consumption equals autonomous consumption $\bar{C}$ plus the marginal propensity to consume $c$ times disposable income $Y_d$
• MPC ($c$) typically ranges from 0.6 to 0.9; MPS = $1 - c$ is the marginal propensity to save
• Multiplier effect: $\frac{1}{1-c}$—shows how initial spending changes amplify through the economy; higher MPC means larger multiplier

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Quick Reference Table

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Self-Check Questions

1. Both utility maximization and cost minimization involve tangency conditions. What two ratios must be equal in each case, and why does this make economic sense?

2. If a Cobb-Douglas production function has $\alpha = 0.3$ and $\beta = 0.8$, what happens to output if both inputs double? What type of returns to scale does this exhibit?

3. A firm finds that $MR > MC$ at its current output level. Should it increase or decrease production? Explain using the profit maximization condition.

4. Compare the substitution effect and income effect for a price increase on a normal good versus an inferior good. In which case might the total effect be ambiguous?

5. If the marginal propensity to consume is 0.75, calculate the spending multiplier. How would an autonomous increase in investment of $100 billion affect equilibrium GDP?