In oligopolistic markets, a firm's optimal strategy depends on the actions of its rivals. We move beyond simple monopoly or perfect competition to model this interdependence. The three canonical models—Cournot, Bertrand, and Stackelberg—offer different perspectives on this strategic chess match, distinguished primarily by the variable firms choose and the timing of their decisions.

The Cournot model assumes firms choose output levels, letting the market determine the price. Each firm's profit maximization problem is solved by taking the derivative of its profit function with respect to its own quantity, holding the rival's quantity constant. This yields a reaction function, $q_1^* = R_1(q_2)$, showing the optimal output for firm 1 for any output chosen by firm 2. The Nash equilibrium occurs where these reaction functions intersect.

In contrast, the Bertrand model posits that firms compete on price. If products are homogeneous, any firm charging a price higher than its competitor's gets zero sales. This creates a powerful incentive to undercut, leading to a price war that ends only when price equals marginal cost. The outcome is allocatively efficient, a striking result for a market with only two firms. Product differentiation is a common way to escape this paradox in the real world.

The Stackelberg model introduces sequential decision-making. The leader firm anticipates the follower's reaction function and incorporates it directly into its own profit maximization problem, effectively choosing a point on the follower's reaction curve that maximizes its own profit.

When the price of a good changes, a consumer's purchasing decision is altered through two distinct channels: the substitution effect and the income effect. The substitution effect captures the change in consumption due to the change in the relative prices of goods, holding utility constant. The income effect captures the change due to the shift in real purchasing power.

The Slutsky equation precisely decomposes the total effect of a price change into these two components.

Graphically, we can visualize this decomposition. The substitution effect is the movement along the original indifference curve to a point where its slope equals the slope of the new budget line. The income effect is the shift from that intermediate point to the final consumption bundle on the new indifference curve.

Cost functions and production functions are two sides of the same coin. For any given production technology, there exists a unique minimum cost function that yields a certain level of output. This concept is known as duality. Analyzing the firm's behavior through its cost function is often more convenient than working directly with the production function.

The shape of a firm's cost curves is a direct reflection of the properties of its production function. For instance, the law of diminishing marginal returns is what gives the marginal cost curve its upward slope.

A firm's long-run expansion path traces out the cost-minimizing combinations of inputs for all levels of output. It connects the tangency points between isoquants and isocost lines. The slope of this path provides information about the firm's input usage as it scales production. For a homothetic production function, the expansion path is a straight line through the origin, meaning the optimal input ratio remains constant as output changes.

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These models provide the analytical tools to dissect complex market behaviors and firm-level decisions, forming the core of modern microeconomic analysis.