Homework 1 — Comprehensive Study Notes (Feb 11, 2026) Summary & Study Notes

🧮 Question 1 — Demand and Price Elasticity

Demand function: $Q_D = 10{,}000 - 10P$. With prices $P_1 = 500$ and $P_2 = 400$ we get quantities:

$Q_1 = 10{,}000 - 10\cdot 500 = 5{,}000$ and $Q_2 = 10{,}000 - 10\cdot 400 = 6{,}000$.

Arc (midpoint) price elasticity of demand:

$E = \dfrac{\Delta Q / \left(\frac{Q_1 + Q_2}{2}\right)}{\Delta P / \left(\frac{P_1 + P_2}{2}\right)} = \dfrac{1{,}000 / \left(\frac{5{,}000 + 6{,}000}{2}\right)}{-100 / \left(\frac{500 + 400}{2}\right)} = -0.82$.

Interpretation: a 1% increase in rent reduces housing demand by 0.82%. Because |E| < 1, demand is inelastic.

🩺 Question 2 — Health, Job Loss, and the MEC Curve

Key concept — MEC curve: the Marginal Efficiency of Capital (MEC) curve is used here to illustrate optimal health choices. Optimal health is denoted $H^*$ and shifts when returns or constraints change.

Reasons health might increase after job loss (sample mechanisms):

• More time available: Job loss can increase available time to invest in health (less $T_{work}$), so optimal health may rise because the time constraint eases.

• Lower interest rates: If the rate of return on other investments falls ($r \to r'$), the relative return to investing in health can rise, shifting the MEC and raising the optimal health level from $H^$H∗ to $\tilde{H}^$.

Reasons health might decrease after job loss (sample mechanisms):

• Tighter budget constraint: Job loss reduces income, tightening the budget and lowering resources available for health investment, which can reduce optimal health.

• Allostatic load (stress) hypothesis: Increased stress raises the marginal cost of maintaining health (modeled as an increase in a stress parameter such as $\gamma$), shifting the MEC so optimal health falls from $H^$H∗ to $\tilde{H}^$.

In short, whether health increases or decreases depends on the relative strength of time availability, income effects, returns to health investment, and stress effects.

🕒 Question 3 — Labor–Leisure Diagram and the Gym Example

Setup and time endowment: With 10 hours for sleep, the individual has 14 hours available for work ($T_W$), leisure ($T_L$) and health-related activities ($T_H$). The time constraint is $T_H + T_W + T_L = 14$.

Example with gym attendance: if one hour is spent at the gym then $T_H = 1$. If the wage is $w = 15$ and the person uses the remaining 13 hours for work, income is $15 \cdot 13 = 195$ and leisure $T_L = 0$ in that extreme point. The budget line can be drawn either in time units or income units since income = $w \cdot T_W$.

Effect of the gym closing: if the gym closes, assume the gym previously reduced time lost to illness by at least an hour. Then $T_H$ falls to 0 and the effective productive time available may decline (because sickness increases), which tightens the feasible set. Graphically, this is represented as an inward shift of the budget constraint (less effective time/income for any given leisure choice). The gym closure therefore reduces the attainable combinations of work, leisure and health.

Key terms: budget constraint, time endowment, gym closure and how changes in nonmarket activities can alter time available for work and leisure.

🎲 Question 4 — Expected Value and Risk Aversion

Expected value calculation: Consider a lottery that pays $90$ with probability $\frac{1}{3}$ and pays $-42$ with probability $\frac{2}{3}$. The expected value is:

$E[Lottery] = \frac{1}{3}\cdot 90 + \frac{2}{3}\cdot (-42) = 30 - 28 = 2$.

Interpretation: the lottery has a positive expected value of </span>2$.

Risk aversion: If the individual is indifferent between taking this lottery (positive EV) and walking away, that reveals risk aversion: their utility function is concave, so the disutility from potential losses outweighs the benefit of the positive expected dollar payoff. In other words, they prefer the certain alternative (walk away) to the risky lottery despite a positive expected monetary value.