Study sheet: Modeling Monetary Economies

📋 Course Outline

1. Overlapping Generations Framework
2. Endowments and Nonstorable Goods
3. Intertemporal Preferences
4. Preference Consistency and Initial Old
5. Planner Feasibility Constraints
6. Golden Rule Allocation

📖 1. Overlapping Generations Framework

🔑 Key Concepts & Definitions

• Overlapping Generations Economy : Paul Samuelson, 1958 — an economy in which young and old individuals coexist in the same period

★ Must-know

• Individuals live for two periods, first as young and then as old.

• For every period t ≥ 1, Nt agents are born, become young in period t, and become old in period t+1.

Further detail

• At t = 1, N0 initial old individuals live for only one period.

💡 Memory Hook

A red young generation overlaps a black old generation in every period.

📖 2. Endowments and Nonstorable Goods

🔑 Key Concepts & Definitions

• Endowment : Each individual receives y consumption goods when young and nothing when old

📝 Essential Points

• In each period, the economy contains Nt young people and Nt−1 old people.

📌 The single consumption good cannot be stored from one period to the next.

• Because goods cannot be stored and individuals want consumption in both periods, trade between generations is necessary.

💡 Memory Hook

Young people receive goods, whereas old people receive nothing.

📖 3. Intertemporal Preferences

🔑 Key Concepts & Definitions

• Consumption Bundle : an individual's consumption when young, c1,t, and when old, c2,t+1
• Indifference Curve : connects consumption bundles that yield the same utility to an individual

📝 Essential Points

📐 Formula — The marginal rate of substitution equals the absolute value of the indifference curve's slope: $MRS = \frac{\partial U(c_1,c_2)/\partial c_1}{\partial U(c_1,c_2)/\partial c_2}$.

📌 As c1 increases along the indifference curve, the curve becomes flatter and the marginal rate of substitution diminishes.

💡 Memory Hook

More young consumption requires willingness to give up old consumption, captured by the MRS.

📖 4. Preference Consistency and Initial Old

★ Must-know

📌 If an agent prefers bundle B to A and bundle C to B, transitivity requires the agent to prefer C to A.

📌 Transitivity implies that indifference curves cannot cross.

• The initial old live and consume only in the initial period and maximize consumption subject to their endowments.

Further detail

📌 If bundles B and A are equally preferred and bundles C and B are equally preferred, transitivity requires bundles A and C to be equally preferred.

💡 Memory Hook

Future generations choose bundles across two periods, whereas the initial old consume only in the initial period.

📖 5. Planner Feasibility Constraints

★ Must-know

📌 The centralized solution uses a benevolent planner, whereas the decentralized solution uses trade with money.

• The planner's available resources in period t equal $N_t y$.

📌 Under equity, every member of generation t receives the same allocation, so total young consumption is Nt c1,t and total old consumption is Nt−1 c2,t.

📐 Formula — Total consumption is feasible when $N_t c_{1,t}+N_{t-1}c_{2,t}\leq N_t y$.

Further detail

📐 Formula — With a constant population, Nt−1 = Nt = N and feasibility becomes $c_{1,t}+c_{2,t}\leq y$.

💡 Memory Hook

Resources → equal allocations → aggregate feasibility → stationary allocation.

📖 6. Golden Rule Allocation

🔑 Key Concepts & Definitions

• Stationary Allocation : A stationary allocation gives every generation the same lifetime consumption, so c1,t = c1 and c2,t+1 = c2 for every date.
• Golden Rule Allocation : the stationary feasible allocation that maximizes the welfare of future generations, measured by U(c1,c2)

★ Must-know

📐 Formula — For a stationary allocation with constant population, feasibility is $c_1+c_2\leq y$.

• The initial-old optimal allocation is the feasible allocation that maximizes consumption for the initial old.

Further detail

📌 The golden rule allocation and the initial-old optimal allocation can differ because they maximize welfare for different groups.

💡 Memory Hook

The golden rule maximizes future-generation welfare, whereas the initial-old optimum maximizes initial-old consumption.