本文主要介绍了双曲贴现理论的推导过程与附加结论。

## The origin of the paper

### The characteristic of the hyperbolic discount function:

Dynamically inconsistent preferences.

### Implying

➡️ Implying a motive for consumers to constrain their own future choices.

## Theory of Commitment

### Strotz’s work

#### Achievement

**Costly commitment decisions are commonly observed.**

EG: The saying goes, the saving plans, personal financial management firms, and etc. ➡️ "inability to default is the force of the commitment".

#### Not exhaustive

In general, all illiquid assets provide a form of commitment, though there are sometimes additional reasons that consumers might hold such assets.

Moreover, borrowing against some of these assets is legally treated as an early withdrawal, and hence also subject to penalty.

A less transparent instrument for commitment is an investment in an illiquid asset that generates a steady stream of benefits, but that is hard to sell due yo substantial transactions costs, informational problems, or incomplete markets.

⬆️ **All of the illiquid assets dis- cussed above have the same property as the goose that laid golden eggs. The asset promises to generate substantial benefits in the long run, but these benefits are difficult, if not impossible, to realize immediately. Trying to do so will result in a substantial capital loss.**

#### Leading into a new function

Hyperbolic discount curve.

## Hyperbolic function

### Features

Hyperbolic discount functions are characterized by a relatively high discount rate over short horizons and a relatively low discount rate over long horizons. This discount structure sets up a conflict between today’s preferences, and the preferences that will be held in the future.

### Affect

- This framework predicts that consumption will track income.
- The model explains why consumers have a different propensity to consume out of wealth than they do out of labor income.
- The model explains why Ricardian equivalence should not hold even in an economy characterized by an infinitely lived representative agent.
- The model suggests that financial innovation may have caused the ongoing decline in U. S. savings rates, since financial innovation increases liquidity and eliminates implicit commitment opportunities.
- The model provides a formal framework for considering the proposition that financial market innovation reduces welfare by providing “too much” liquidity.

## The consumption decision based on the commitment

### Golden Egg Model and the Extension

#### Characteristic

- Highly stylized commitment technology that is amenable to an analytic treatment.
- So a current decision to liquidate part or all of an individual’s z holding will generate cash flow that can be consumed no earlier than next period.
- By contrast, agents can always immediately consume their x holdings.

#### Assumption

Consumers may invest in 2 instruments:

- A liquid asset x and an illiquid asset z.
- Instrument z is illiquid in the sense that a sale of this asset has to be initiated one period before the actual proceeds are received.

#### Reduction

Such contracts are susceptible to renegotiation by to- morrow’s self, and in any finite-horizon environment, the contract would unwind. (In the second to last period renegotiation would occur, implying renegotiation in the third to last period, etc.) Second, such contracts are generally unenforceable in the United States. To make such a contract work, tomorrow’s self must be forced to pay the specified funds to the outside agent or be penalized for not doing so (note that the transfer is not in the interest of tomorrow’s self). However, U. S. courts will generally not enforce contracts with a penalty of this kind.

## Hyperbolic Discount Function

### Characteristics

- Hyperbolic discount functions imply discount rates that decline as the discounted event is moved further away in time. Events in the near future are discounted at a higher implicit discount rate than events in the distant future.
- an exponential discount function, $\delta ^t$ is characterized by a constant discount rate, $log(\frac{1}{\delta})$, while the generalized hyperbolic discount function is characterized by an instantaneous discount rate that falls as t rises

### Instantaneous Discount Rate

Was defined in: $ - \frac{f'(\tau)}{f(\tau)} $.

## Equilibrium Strategies

### Problems

Unfortunately, for general interest rate and labor income sequences, it is not possible to use marginal conditions to characterize the equilibrium strategies. This non-marginality property is related to the fact that selves who make choices at least two periods from the end of the game face a nonconvex reduced-form choice set, where the reduced- form choice set is defined as the consumption vectors which are attainable, assuming that all future selves play equilibrium strategies. The nonconvexity in the reduced-form choice set of self T- 2 generates discontinuous equilibrium strategies for self T-2, which in turn generate discontinuities in the equilibrium payoff map of self T- 3. This implies that marginal conditions cannot be used to characterize the equilibrium choices of selves at least three periods from the end of the game.

### Solution:

A Restriction on the Labor Income Process that Eliminates these Problems:

$$ u'(y_t) = \beta \delta^t (\prod_{t=1}^{ \tau } R_{t+i} ) u' (y_{t+\tau}) \quad \forall t,\tau \ge 1 $$

### Main Part

#### Assumption and Definition

Before characterizing the equilibria of the game, it is helpful to introduce the following definitions. First, we will say that a joint strategy, s, is resource exhausting if $s|h_{T+1}

$, is characterized by $ z_T = x_t = 0 $, for all feasible $h_{T-1}$. Second, we will say that a sequence of feasible consumption/savings actions, $\left { c_\hat{t}, x_\hat{t}, z_\hat{t}, ... , c_T, x_T, z_T \right } $ satisfies P1-P4 if $\forall t \ge \hat{t}$,

$$ P1: u'(c_t) \ge \max_{\tau \in (1,...,T-t)} \beta \delta^\tau (\prod_{t=1}^{\tau} R_{t+i})u'(c_{t+\tau}) $$

$$ P2: u'(c_t) > \max_{\tau \in (1,...,T-t)} \beta \delta^\tau (\prod_{t=1}^{\tau} R_{t+i})u'(c_{t+\tau}) \Longrightarrow c_t = y_t + R_tx_{t-1} $$

$$ P3: u'(c_t) < \max_{\tau \in (1,...,T-t-1)} \beta \delta^\tau (\prod_{t=1}^{\tau} R_{t+i})u'(c_{t+1+\tau}) \Longrightarrow x_t = 0 $$

$$ P4: u'(c_t) > \max_{\tau \in (1,...,T-t-1)} \beta \delta^\tau (\prod_{t=1}^{\tau} R_{t+i})u'(c_{t+1+\tau}) \Longrightarrow z_t = 0 $$

Finally, we will say that a joint strategy $ s \in S $ satisfies P1-P4 if for an feasible history $ h_{\hat{t}}, s|h_{\hat{t}} $ satisfies P1-P4.

#### Theorem 1

Fix any T-period consumption game with exogenous variables satisfying A1. There exists a unique resource- exhausting joint strategy, $s \in S$, that satisfies P1–P4, and this strategy is the unique subgame perfect equilibrium strategy of this game.

➡️ Hence, no wealth goes unconsumed in equilibrium.

## Analysis based on the Golden Egg Model

- Comovement of Consumption and Income
- Aggregate Saving.
- Assets-Specific MPCs
- Ricardian Equivalence
- Declining Savings Rates in the 1980s
- Welfare Analysis of Financial Innovation

## Evaluation and Extensions

### Some black points

- The golden eggs model does not explain how consumers accumulate assets in the first place.
- The anomalous prediction that consumers will always face a binding self- imposed liquidity constraint.
- Some consumers may not need to use external commitment devices (like illiquid assets) to achieve self-control.
- some consumers may have access to an array of “social” commitment devices that are far richer than the simple illiquid asset proposed in this essay.