Fundamental Economic Equations

Why This Matters

Mathematical economics is about understanding the logical structure that connects individual decisions to market outcomes. Every equation here 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 a free-response asking you to "show mathematically" why a firm should change its output.

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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

$Q_d = Q_s$ is the fundamental market-clearing condition. At the equilibrium price $P^*$, the quantity buyers want to purchase exactly equals the quantity sellers want to offer.

To find equilibrium, set the demand function equal to the supply function ($D(P) = S(P)$) and solve for $P$. Then plug that price back into either function to get $Q^*$.

• Comparative statics let you predict how shifts in either curve (from taxes, preference changes, or input costs) alter $P^*$ and $Q^*$. You do this by re-solving the system after modifying the relevant function.

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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. 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 $K$, where $A$ represents total factor productivity (a catch-all for technology and efficiency).

Returns to scale depend entirely on the sum $\alpha + \beta$:

• $\alpha + \beta = 1$: constant returns to scale (double inputs, double output)
• $\alpha + \beta > 1$: increasing returns to scale (double inputs, more than double output)
• $\alpha + \beta < 1$: decreasing returns to scale (double inputs, less than double output)

The marginal products are $MP_L = \alpha \frac{Q}{L}$ and $MP_K = \beta \frac{Q}{K}$. These tell you how much extra output you get from one more unit of labor or capital, and they're essential for input optimization problems.

Cost Minimization

The goal here is to produce a given output level at the lowest possible cost. The optimality condition is:

$\frac{MP_L}{w} = \frac{MP_K}{r}$

This tangency condition says the last dollar spent on labor must yield the same additional output as the last dollar spent on capital. If it didn't, you could reshuffle spending between inputs and produce the same output more cheaply.

• The isocost line $C = wL + rK$ represents all input combinations with the same total cost. Optimal production occurs where this line is tangent to the isoquant (the curve of input combos that produce the same output).
• The expansion path traces optimal input combinations as output changes, and it's how you derive long-run cost curves.

Marginal Revenue and Marginal Cost

• Marginal cost: $MC = \frac{dTC}{dQ}$, the additional cost of producing one more unit. It's typically U-shaped because of diminishing returns.
• Marginal revenue: $MR = \frac{dTR}{dQ}$. For a price-taking firm, $MR = P$. For a firm with market power, $MR = P\left(1 + \frac{1}{\varepsilon_d}\right)$, which is always less than price when demand slopes downward (since $\varepsilon_d < 0$).
• In perfect competition, the MC curve above average variable cost is the firm's supply curve.

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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

The canonical consumer problem is:

$\max U(x, y) \text{ subject to } P_x x + P_y y = M$

where $M$ is income and $P_x$, $P_y$ are prices.

The optimality condition is:

$\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$

This says the marginal utility per dollar must be equal across all goods. If good $x$ gave you more utility per dollar than good $y$, you'd buy more $x$ and less $y$ until the ratios equalized.

The Lagrangian method sets up $\mathcal{L} = U(x,y) + \lambda(M - P_x x - P_y y)$. The multiplier $\lambda$ has a concrete interpretation: it's the marginal utility of income, telling you how much your maximized utility increases if you get one more dollar of budget.

Marginal Rate of Substitution

$MRS_{xy} = \frac{MU_x}{MU_y}$ measures the rate at which a consumer willingly trades good $y$ for good $x$ while staying on the same indifference curve.

• Geometrically, it's the negative of the slope of the indifference curve at any point. Diminishing MRS (the curve gets flatter as you move right) reflects convex preferences, meaning consumers prefer balanced bundles.
• At the optimum: $MRS_{xy} = \frac{P_x}{P_y}$. The subjective trade-off equals the market trade-off. If these weren't equal, the consumer could rearrange purchases and reach a higher indifference curve.

Income and Substitution Effects

When a price changes, two things happen simultaneously:

• Substitution effect: the change in quantity demanded due solely to the relative price change, holding utility constant. This always goes opposite to the price change (price up, quantity down).
• Income effect: the change in quantity demanded due to the change in real purchasing power. The direction depends on whether the good is normal (income effect reinforces substitution effect) or inferior (income effect works against it).

The Slutsky equation decomposes the total effect formally:

$\frac{\partial x}{\partial P_x} = \frac{\partial x^h}{\partial P_x} - x \frac{\partial x}{\partial M}$

where $x^h$ is the Hicksian (compensated) demand. The first term is the substitution effect; the second is the income effect (note the negative sign and the multiplication by $x$, the quantity consumed).

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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.

The decision rule comes in two parts:

1. First-order condition: $MR = MC$. Produce where the revenue from the last unit exactly covers its cost.
2. Second-order condition: $\frac{dMC}{dQ} > \frac{dMR}{dQ}$. This ensures you're at a maximum, not a minimum. Graphically, MC must cut MR from below.

If $MR > MC$ at your current output, you should produce more (each additional unit adds to profit). If $MR < MC$, you should produce less.

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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 different markets and goods.

Elasticity of Demand

Price elasticity of demand is defined as:

$\varepsilon_d = \frac{\% \Delta Q_d}{\% \Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$

The point elasticity formula (the second expression) is what you'll use most in this course. Note that $\varepsilon_d$ is typically negative since demand curves slope downward, but you'll often see it discussed in absolute value.

• Elastic ($|\varepsilon_d| > 1$): quantity is highly responsive to price. A price cut raises total revenue.
• Inelastic ($|\varepsilon_d| < 1$): quantity barely responds. A price cut lowers total revenue.
• Total revenue test: $TR$ is maximized where $|\varepsilon_d| = 1$ (unit elasticity). This is also the point where $MR = 0$.

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Aggregate Behavior

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

Consumption Function

$C = \bar{C} + cY_d$ says that consumption equals autonomous consumption $\bar{C}$ (spending that happens regardless of income) 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. Together they must sum to 1 because every additional dollar of income is either spent or saved.
• The spending multiplier is $\frac{1}{1-c}$. It shows how an initial injection of spending gets amplified as it cycles through the economy. With $c = 0.8$, the multiplier is $\frac{1}{0.2} = 5$, meaning a $1 billion increase in autonomous spending raises equilibrium GDP by $5 billion.

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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?