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Households allocate time between work and leisure to maximize utility subject to a time constraint and budget constraint. Higher wages create a substitution effect (work more) and income effect (work less). Labor supply elasticity depends on which effect dominates. Reservation wage is the minimum wage inducing market work. Individual labor supply curves can bend backward at high wages (income effect dominates) while aggregate labor supply typically slopes upward.
Your prerequisite — household optimization over consumption and savings — established that households maximize utility subject to budget constraints. Labor supply applies the same framework to time. A household has a fixed time endowment each period (say, 168 hours a week) that it divides between market work and leisure, where leisure is everything that is not paid work: sleep, family time, hobbies, home production. The wage rate is the price of leisure — every hour spent not working costs you the wage you could have earned.
This framing lets you apply the income-substitution decomposition you already know. When the wage rises, two forces operate simultaneously. The substitution effect says: leisure is now more expensive relative to consumption goods, so rational households substitute away from leisure toward work — they supply more labor hours. The income effect says: at a higher wage, the household is richer (each hour worked buys more goods), so it demands more of all normal goods including leisure — it works fewer hours. The wage increase triggers both effects at once, and they pull in opposite directions. At low wage levels, the substitution effect typically dominates and labor supply increases with the wage. At high wage levels, workers may already have abundant consumption and place increasing value on their time, so the income effect begins to dominate.
The result is the backward-bending labor supply curve: as the wage rises from very low levels, hours worked increase; beyond a turning point, further wage increases reduce hours supplied as workers "buy" more leisure with their higher income. This explains real-world patterns like professionals working fewer total hours after a windfall, or high-income workers reducing hours when given pay raises, even though it seems counterintuitive. The reservation wage — the minimum wage that induces someone to enter the labor market at all — is the wage at which the utility from working just equals the utility of not working; below it, the person stays out of the labor market entirely.
Market labor supply aggregates individual decisions and typically slopes upward because workers enter the market at different reservation wages. As the wage rises, workers who were indifferent between working and not working are drawn in, more than offsetting the backward-bending tendency of high-income workers already in the market. This population effect — the extensive margin of new entrants — dominates the intensive margin of hours adjustments, keeping aggregate labor supply upward sloping over the ranges most relevant for policy analysis. Minimum wage debates, labor force participation rates, and responses to tax changes all hinge on understanding which margin is operating in a given context.