ECON5001 · Microeconomic Theory
Intertemporal Choice, Labour-Leisure & Uncertainty
This is the pay-off chapter for consumer theory: one budget line, three famous applications. ECON5001 Week 5 reuses the same budget-and-tangency machinery to ask three new questions — how much to save (the interest rate is the price of consuming now versus later), how much to work (the wage is the price of leisure), and how to handle risk (you now choose over states of the world, ranking prospects by expected utility rather than expected dollars). Each looks like a separate topic, but every one is the same optimisation with the axes and the “price” relabelled, which is exactly why the exam can drill all three from one method. The marks are in setting up the budget, imposing the optimum, and answering the actual question — saving, hours, or whether to insure.
What this chapter covers
- 011. Present & future value — FV = m(1+r), PV = m/(1+r); a dollar later is worth less than a dollar today
- 022. Intertemporal budget constraint — (1+r)c₁ + c₂ = (1+r)m₁ + m₂; slope −(1+r), pivots about the endowment
- 033. Saver vs borrower — c₁ < m₁ saves, c₁ > m₁ borrows; optimum MRS = 1+r; a higher r helps savers
- 044. Consumption smoothing — convex preferences spread consumption; with min{} preferences set c₁ = c₂
- 055. Labour-leisure choice — pC = M + w(T − R); the wage is the price of leisure, real wage w/p = slope
- 066. Backward-bending labour supply — substitution effect vs income effect of a wage rise
- 077. Expected utility (von Neumann–Morgenstern) — EU = Σ πᵢ u(xᵢ), not expected money EV = Σ πᵢ xᵢ
- 088. Risk, certainty equivalent & insurance — risk premium = EV − CE; full insurance at a fair premium
Expected utility, certainty equivalent & the insurance decision
- +2(a) Expected utility. Average utility across the two states: EU = 0.5·√64 + 0.5·√100 = 0.5(8) + 0.5(10) = 4 + 5 = 9.
- +1Certainty equivalent. CE is the sure wealth with the same utility: u(CE) = EU ⇒ √CE = 9 ⇒ CE = 81.
- +2(b) Maximum willingness to pay. Versus her no-loss wealth of 100, she would pay up to 100 − CE = 100 − 81 = 19 for guaranteed full protection.
- +1(c) Expected wealth. E[W] = 0.5(64) + 0.5(100) = 82, so the actuarially fair premium equals the expected loss = 0.5(36) = 18.
- +2Risk premium & decision. RP = E[W] − CE = 82 − 81 = 1. Because the fair premium (18) is below her willingness to pay (19), a risk-averse agent strictly buys and fully insures.
Key terms
- Present value / future value
- Discounting at interest rate r. Future value pushes money forward in time: FV = m(1+r). Present value brings money back: PV = m/(1+r). A dollar next period is worth less than a dollar today because today's dollar can be lent at r. The factor (1+r) is the conversion rate between the two periods.
- Intertemporal budget constraint
- The two-period budget over consumption now (c₁) and later (c₂). Future-value form: (1+r)c₁ + c₂ = (1+r)m₁ + m₂. Present-value form: c₁ + c₂/(1+r) = m₁ + m₂/(1+r). Both describe the same line, with slope −(1+r), that passes through the endowment (m₁, m₂). The interior optimum sets MRS = 1+r.
- Saver vs borrower
- Choosing c₁ < m₁ means consuming less than current income and carrying purchasing power forward — a saver/lender who earns r. Choosing c₁ > m₁ means pulling future income forward — a borrower who pays r. A rise in r pivots the budget about the endowment (steeper), helping savers and possibly turning a borrower into a saver.
- Consumption smoothing
- Convex intertemporal preferences make the consumer prefer to spread consumption across periods rather than concentrate it. In the extreme of perfect complements, U = min{c₁, c₂}, the optimum sits on the 45° line, so you simply set c₁ = c₂ and solve a single budget equation.
- Labour-leisure choice
- Splitting a time endowment T between leisure R and work L = T − R. The budget is pC = M + w(T − R), where M is non-labour income and w is the wage; the budget slope (in consumption–leisure space) is −w/p, the real wage. Because an hour of leisure costs the wage you forgo, the wage is the price of leisure. A wage rise can produce a backward-bending supply curve when the income effect outweighs the substitution effect.
- Expected utility (von Neumann–Morgenstern)
- A rational agent facing risk maximises expected utility EU = Σ πᵢ u(xᵢ), the probability-weighted utility of outcomes — not the expected money value EV = Σ πᵢ xᵢ. The shape of u(W) encodes the attitude to risk: concave means risk-averse, linear means risk-neutral, convex means risk-loving.
- Certainty equivalent & risk premium
- The certainty equivalent CE is the sure wealth giving the same utility as the gamble: u(CE) = EU. The risk premium RP = EV − CE is the maximum a risk-averse agent will pay to avoid the risk. For a concave u the chord lies below the curve, so EU is less than u(EV), CE is below EV, and the risk premium is positive.
- Actuarially fair insurance
- A premium is actuarially fair when the premium per dollar of cover equals the loss probability π (so the premium equals the expected loss). Faced with a fair premium, a risk-averse agent fully insures and equalises wealth across states (the optimum lands on the 45° certainty line). At an unfair premium she still buys some cover but generally under-insures.
Intertemporal Choice, Labour-Leisure & Uncertainty FAQ
Do I multiply or divide by (1+r) when discounting?
It depends on the direction in time. To move money forward (today → next period) you multiply: FV = m(1+r). To move money back (next period → today) you divide: PV = m/(1+r). The single most common slip in this chapter is dividing when you should multiply or vice versa, so write down which way in time you are travelling before you reach for the factor. The same (1+r) is the slope of the intertemporal budget line, so it appears everywhere in the two-period problem.
What is the difference between a saver and a borrower, and how does a higher interest rate affect them?
A saver consumes less than current income (c₁ < m₁) and lends the rest, earning r; a borrower consumes more (c₁ > m₁) and pays r. A rise in r is a price change for future consumption, so it pivots the budget about the endowment (m₁, m₂) rather than shifting it in parallel — the no-trade point is always affordable. A higher r makes a saver better off (and still a saver); a borrower is worse off and may switch to saving.
Why maximise expected utility instead of expected money value?
Because rational choice under risk is about the utility of wealth, not wealth itself. A risk-averse agent has a concave utility function (for example √W or ln W), so the marginal utility of money falls as wealth rises. That makes the expected utility of a gamble lower than the utility of its expected value, so she will refuse a fair bet and pay to avoid risk. Ranking by expected money value would ignore this curvature and wrongly predict indifference to risk.
Why can a labour-supply curve bend backward?
A wage rise has two opposing effects. The substitution effect makes leisure more expensive, so the worker takes less leisure and works more. The income effect makes the worker richer, so — if leisure is a normal good — she takes more leisure and works less. At low wages the substitution effect dominates and hours rise with the wage; past a critical wage the income effect dominates and hours fall, so the supply curve bends backward. When asked for the sign of the response, name both effects before committing.
Does a risk-averse person always fully insure?
Only when the premium is actuarially fair (the premium per dollar of cover equals the loss probability). At a fair premium a risk-averse agent fully insures and equalises wealth across all states — the optimum sits on the 45° certainty line. At an unfair premium (loaded above the fair rate) she still buys some cover but generally under-insures, stopping short of full coverage. Always state which premium type the question gives before asserting full insurance.
How do I compute a risk premium or willingness-to-pay for insurance?
Work in utility space, not by averaging dollars. First find expected utility EU = Σ πᵢ u(xᵢ). Then invert the utility function to get the certainty equivalent from u(CE) = EU (so CE = EU² for u = √W). The risk premium is EV − CE, and the maximum willingness to pay for full protection is the no-loss wealth minus CE. If the fair premium is below that willingness to pay, the agent buys.
Exam move
Treat Chapter 5 as one engine with three sets of labels rather than three separate topics, because the exam drills all of them from the same routine. Step one is always to write the budget: c₁ + c₂/(1+r) = m₁ + m₂/(1+r) for intertemporal choice, pC = M + w(T − R) for labour-leisure, or a list of states with their probabilities for uncertainty. Step two imposes the optimum: MRS = 1+r (or = w/p) for the smooth case, c₁ = c₂ for perfect-complement smoothing, or compute EU and invert to the certainty equivalent under uncertainty. Step three answers the question actually asked — how much she saves, how many hours she works, or whether she insures — and states the sign rule you used. Drill the recurring traps until they are automatic: discount future money by dividing by (1+r) (never multiply); remember an interest-rate change pivots the budget about the endowment; maximise expected utility, not expected dollars; recall that labour supply can bend backward; and only claim full insurance when the premium is actuarially fair. Redo each worked example with your own numbers after reading the method, and practise against the clock so the setup is reflexive.
