5. The representative household and competitive equilibrium
Introduction
In the household block, a representative agent chooses consumption, investment and time devoted to paid work. Leisure provides direct utility but forgoing an hour of work sacrifices wages that could finance consumption. Saving provides tomorrow's productive capital but requires resources to be withheld from consumption today.
Dynamic optimization generates two economically distinct equilibrium conditions. The Euler equation balances the marginal utility of consuming today against the discounted expected benefit of investment and future consumption. The intratemporal labor condition compares the marginal utility of leisure with the wage measured in utility units.
Firm optimality, household optimality, market clearing and the economy-wide resource constraint are all needed for competitive equilibrium. A condition derived for one agent is not sufficient: all desired factor purchases and supplies must agree, and aggregate expenditure cannot exceed production.
Integrated theoretical explanation
The representative household begins with a stock of installed capital and a normalized time endowment of one unit each period. Time is divided between labor nₜ and leisure ℓₜ, so nₜ + ℓₜ = 1. Working more hours generates wage income but removes leisure; retaining more current output for consumption reduces the resources that can finance investment. The household's decisions therefore combine an intratemporal trade-off between consumption and leisure and an intertemporal trade-off between consumption today and consumption in the future.
Lifetime utility takes the form E₀Σβᵗu(cₜ,ℓₜ), where β between zero and one discounts future utility and E₀ represents expectations given information available at the initial date. A general family of consumption–leisure preferences can be specialized to logarithmic consumption and linear utility from leisure, u(c,ℓ) = ln c + ϕℓ. Under this specification, the marginal utility of consumption equals 1/c and the marginal utility of leisure is the constant ϕ. Concavity in consumption makes a more evenly distributed consumption path preferable to large fluctuations with the same average level.
The household faces a budget restriction that combines the wage bill, the rental income earned on installed capital, distributed profits and undepreciated capital. Writing next-period productive capital on the spending side, cₜ + kₜ = wₜnₜ + rₜkₜ₋₁ + πₜ + (1 − δ)kₜ₋₁, expresses the use of all available resources when the optimum is interior. This is equivalent to accounting for investment separately with kₜ = (1 − δ)kₜ₋₁ + iₜ. The depreciation rate δ measures the fraction of old capital physically lost between periods.
The Euler equation is the condition for optimal saving. Sacrificing one unit of consumption in period t gives up current marginal utility 1/cₜ but increases capital available for the next period. That capital produces rental income rₜ₊₁ and retains the undepreciated fraction 1 − δ. The expected future payoff must therefore be discounted and weighted by marginal utility at t+1. At an interior optimum, 1/cₜ = βEₜ[(rₜ₊₁ + 1 − δ)/cₜ₊₁]. If the expected return rises, households generally have an incentive to save more, although the total response also depends on wealth and future income expectations.
The intratemporal first-order condition compares the utility lost from less leisure with the utility gained from wage-financed consumption. Under the specified utility function, ϕ = wₜ/cₜ, or equivalently ϕcₜ = wₜ. Combining this with competitive labor demand yields ϕcₜ = (1 − α)yₜ/nₜ. The equation does not say that households supply a fixed number of hours: it makes labor supply jointly dependent on consumption, productivity and the equilibrium wage. A change in labor productivity alters the wage opportunity cost of leisure and therefore the desired allocation of time.
Consumption smoothing and labor adjustment interact following a technology shock. Higher current productivity provides more goods and raises the potential return on capital. Forward-looking households need not consume all extra production immediately: saving part of it supports consumption in later periods when the technology shock has diminished. At the same time, higher real wages affect the relative attractiveness of leisure and work. A consistent equilibrium determines both margins together; one cannot infer the labor response from the firm's marginal-product equation alone or the consumption response from a budget identity alone.
A competitive equilibrium is a sequence of allocations, factor prices and expectations for which households maximize utility subject to their constraints, firms maximize profits, markets clear, capital follows its accumulation law and productivity follows its exogenous process. Under the closed-economy benchmark, the goods market requires yₜ = cₜ + iₜ. The capital-service and labor markets equate household supply with firm demand. Inserting the competitive factor-price equations into household optimality reduces the model to a compact set of dynamic equations for y, c, n, k and z, which is the starting point for calibration and solution.
Formal framework and essential relationships
The household utility is specialized to logarithmic consumption and linear leisure. The discount factor β determines the weight on future utility; ϕ governs the marginal utility of leisure. Each time endowment is normalized to one.
Consumption plus investment in next period’s productive capital is financed by wages, capital rental income, profits and undepreciated old capital.
Euler/Lucas asset-pricing equation. Investing one resource unit sacrifices 1/cₜ units of marginal utility today and yields a random gross payoff tomorrow. Equality characterizes an interior optimum.
At the intratemporal optimum, the marginal disutility from additional working time balances the real wage valued at the marginal utility of consumption; the second equality clears the labor market.
In the benchmark closed economy with neither public expenditure nor external trade, all output is allocated to consumption or gross investment. Combining this identity with capital accumulation yields the aggregate resource constraint.
Explanations, derivations and empirical applications
Endowments
Household optimization allocates resources across consumption, saving and leisure. The Euler equation and the labor–leisure condition are necessary restrictions that must hold simultaneously with market clearing and the feasible resource allocation.
Households enter the first period with installed productive capital and receive one unit of time in each period. They divide time between working hours and leisure, imposing 1 = nₜ + ℓₜ. Labor produces wage income, whereas leisure yields direct satisfaction but no production. The inherited capital stock finances rental income and provides a state variable linking past investment choices to current consumption opportunities.
Preferences
The preference function must make consumption and leisure desirable and satisfy concavity restrictions. For the non-logarithmic form, restrictions on v and its derivatives depend on σ; marginal-sign conditions alone do not establish joint concavity. The benchmark specializes to:
Why This Utility Function?
Balanced-growth-compatible preferences restrict how consumption and leisure enter utility. Logarithmic consumption with separable leisure is a useful special case: it permits consumption and real wages to grow while hours remain stationary under suitable technology and equilibrium conditions. It is a member of a broader family of preferences, not the uniquely possible utility function.
Rational Expectations
Rational expectations mean that decisions are based on the conditional probability distribution implied by the model and all information available at the decision date. They do not imply that agents know future innovations. Rather, expectations are mathematically consistent with the economy’s stated technology process, equilibrium prices and decision rules. This requirement is indispensable for interpreting the Euler equation in a stochastic environment.
Household’s Optimization Problem
The household takes initial capital as given. Feasibility, nonnegativity and an appropriate transversality condition complement the interior first-order conditions.
First-order condition for capital
Capital chosen at t appears as a cost at t and earns rental income plus an undepreciated remainder at t+1. Differentiating both appearances gives:
Only current information is available when the future payoff is evaluated.
Optimal Capital Supply
The choice of next-period capital equates the utility cost of sacrificing consumption today with the discounted expected utility value of the gross return tomorrow. That return includes the capital rental payment and the undepreciated remainder of the asset. With logarithmic consumption utility, the resulting Euler equation can be expressed as 1/cₜ = βEₜ[(rₜ₊₁ + 1 − δ)/cₜ₊₁].
Household Optimization: Intuition
The marginal utility cost of less current consumption equals the discounted expected utility benefit of future capital returns. The future rental return depends on the future technology realization and allocation:
If the discounted expected marginal utility benefit of saving exceeds its current utility cost, the household can improve utility by investing more. If the inequality is reversed, it can improve utility by consuming more now. An interior optimum equates the two sides of the Euler condition.
Abstracting from expectations for a second, we can uncover a little more intuition.
In a deterministic environment, intertemporal optimality equates willingness to exchange current for future consumption to the feasible gross return:
The left side is the intertemporal marginal rate of substitution; the right side is the marginal rate of transformation. Under uncertainty, the Euler equation uses the conditional expected utility-weighted return.
Consumption Smoothing: Intuition
Smoothing follows from concave utility, not from a rule that consumption must literally stay constant. Expected increases in lifetime resources can raise consumption immediately, while the stock of capital absorbs much of the short-run variation in saving.
Concavity creates a preference for smoother consumption, conditional on lifetime resources, relative prices, uncertainty and borrowing opportunities. It does not require consumption to be constant.
A favorable productivity realization, zₜ above its reference level of one, raises output for given inputs. If labor does not fall enough to offset this effect, the additional output must be absorbed by consumption, investment or both.
As agents want to smooth consumption, they want to spread the increase in output over multiple periods. In other words, they don’t want to consume the entire increase ∆ct < ∆yt; Hence, agents want to invest a portion of the increased output ∆it > 0, as investment postpones consumption to the future.
Consumption Smoothing: Propagation
An impulse response is a controlled thought experiment rather than a historical forecast. The only new shock arrives initially; subsequent responses reflect the model's internal propagation. Capital is predetermined at the start of a period, while employment and consumption may adjust contemporaneously. Simulation repeats this mechanism over many random shocks to generate model-implied moments.
1 Positive technology shock increases output ∆yt; 2 Due to their desire to smooth consumption, investment increases ∆it > 0; 3 This increases the capital stock for tomorrow ∆kt = ∆it; 4 and so output in the future increases; 5 return to step 1).
But: we haven’t shown that labor nt reacts such that output indeed increases in response to a positive technology shock (usually it does, though).
Optimal Labor Supply
The consumption–leisure condition translates a preference trade-off into a market equation. Under logarithmic consumption and linear leisure it becomes ϕcₜ = wₜ, which must be solved jointly with the firm’s marginal product of labor.
Working an additional unit of time reduces leisure by one unit and increases the household’s available consumption by the real wage. With utility ln c + ϕℓ, the marginal utility loss from less leisure is ϕ and the consumption gain has utility value w/c. The intratemporal optimum therefore satisfies ϕcₜ = wₜ. Market clearing then replaces wₜ by the marginal product of labor.
Household Optimization: Intuition
Working one additional unit of time trades off lost leisure against wage-financed consumption. At an interior optimum:
If ul (ct, lt) < uc (ct, lt) · wt the utility gain from additional consumption financed through more work is greater than the utility loss of forgone leisure. Therefore, the household can increase its level of utility by working more and consuming from the additional income.
If however ul (ct, lt) > uc (ct, lt) · wt the utility gain from additional leisure is greater than the utility loss of forgone consumption financed through more work. Therefore, the household can increase its level of utility by working less and enjoying more leisure. So as long as the equality does not hold, it is possible for the household to increase its welfare by adjusting its labor-leisure choice.
The marginal rate of substitution between leisure and consumption equals the real wage, the market’s marginal rate of transformation between the two.
Labor Supply: Propagation
A higher wage creates opposing effects on labor supply. The substitution effect makes leisure more expensive and encourages work; the income effect makes the household wealthier and can increase desired leisure. Their balance depends on preferences, persistence and equilibrium prices. Under the benchmark calibration, hours increase after a positive technology innovation.
If households respond to wt ↑with more labor nt ↑, then the effect of technology shocks increases: Output increases further than with the technology shock alone (amplification). This output increase is only partially consumed: investment increase and so forth following from the consumption smoothing motive.
Market Clearing
Goods, labor and capital markets must clear simultaneously. Total production is absorbed by consumption and investment, labor supplied equals labor hired, and the capital stock rented to firms is the stock owned by households. When firm profits are distributed back to their owners, household budgets are consistent with the aggregate resource identity. These restrictions convert individual optimal decisions into a competitive equilibrium.
Equilibrium
An equilibrium is a situation in which (A) households maximize, (B) firms maximize, (C) markets clear, (D) households form expectations rationally.
Market Clearing I: Labor Market
Market Clearing II: Capital Market
Resource Constraint
Under constant returns and competitive pricing, profits are zero and factor payments exhaust production. Monotonic preferences make the household’s resource constraint bind at equilibrium.
Summary
The nonlinear equilibrium system consists of the Euler condition, labor-market condition, resource constraint, production function and technology process:
Synthesis
Consumption, saving and labor decisions derive from household optimality. Equilibrium requires jointly satisfied Euler, labor, production and resource conditions.