Be With You

Author

Affiliation

Hirotaka Fukui

Kobe University, Graduate School of Economics

Modified

July 14, 2026

생각에 생각이 많아져

엉켜버린 실타래 같지만

너를 바라보는 이 순간의 느낌은

너무나 행복한 걸

Be With You - fromis_9

No-Ponzi Game 条件（NPG）と横断性条件（TVC）の違い

Starting Point: Household’s Budget Constraint

A representative household faces the following budget constraint:

\[ b_{t+1} = (1+r_{t})b_{t} + y_{t} - c_{t} \]

This expression states that next period’s bond holdings are equal to the current assets carried forward with interest, plus current income, minus current consumption.

By iterating this constraint forward indefinitely:

\[ b_{t} = \sum^{\infty}_{j =0}\Biggl( \prod^{j-1}_{i = 0} \frac{1}{1+r_{t+i}} \Biggr) (y_{t+j} - c_{t+j}) + \lim_{j \rightarrow \infty} \Biggl(\prod^{j-1}_{i=0} \frac{1}{1+r_{t+i}} \Biggr) b_{t+j} \]

No-Ponzi Game (NPG) Condition

A household cannot roll over its debt indefinitely by borrowing to finance both principal and interest. This restriction is necessary for the intertemporal budget constraint to hold and for the model to remain internally consistent. It rules out explosive debt paths that would imply an unbounded present value of liabilities. Accordingly, it functions as a feasibility condition of the model:

\[ \lim_{j \rightarrow \infty} \Biggl(\prod^{j-1}_{i=0} \frac{1}{1+r_{t+i}} \Biggr) b_{t+j} = 0 \]

Transversality Condition (TVC)

On the other hand, the condition derived from the household’s optimization problem is:

\[ \lim_{t \rightarrow \infty} \beta^t \lambda_{t} b_{t+1} = 0 \]

This is the transversality condition, which ensures that the household does not find it optimal to accumulate unbounded debt or assets over the infinite horizon. It rules out explosive paths that would violate optimality, and it guarantees that the chosen consumption-saving plan satisfies the necessary boundary condition of the dynamic optimization problem.

Mathematical relationship between two conditions

From the first-order condition:

\[ \beta^t \lambda_{t} = \beta^{t+1} \lambda_{t+1} (1+r_{t})= \beta^{t+1} u'(c_{t+1}) (1+r_{t}) \]

Substituting this expression into the transversality condition yields

\[ \lim_{t \rightarrow \infty} \beta^{t+1} u'(c_{t+1}) (1+r_{t}) b_{t+1}= 0 \]

なぜ transversality condition が必要なのか

有限期間問題と無限期間問題の違い

横断性条件は単なる「終端条件」ではなく、無限遠に残された状態変数・資産・資本・資源の価値をどう扱うかという、無限期間問題に固有の条件として位置づけられます。

Halkin(1974) は古典的な最適制御問題では終端時点があらかじめ与えられている場合でもそうでない場合でも、常に実数であると説明しています。これに対して経済学では終端時点が $+\infty$ になる無限期間問題を扱うため、数学的困難が生じると述べています。

有限期間問題では時間区間を $I = [0, T]$ と置き、状態変数を $x(t)$, 制御変数を $u(t)$ とします。データとして閉集合 $Q \subset \mathbb R^r$, 関数 $(f, L): \mathbb R^n \times Q \times I \rightarrow \mathbb R^n \times \mathbb R$ を考えます。ここで $f$ は状態方程式、$L$ は瞬時利得とします。有限期間の最適制御問題は抽象的には

\[ \max_{u(\cdot)} \int^T_0 L(x(t), u(t), t) dt \]

subject to

\[ \dot(x)(t)=f(x(t), u(t), t), \quad u(t) \in Q, \quad x(0)=x_0 \]

という形で理解できます。重要な点は有限期間問題には終端条件の違いによって、少なくとも2つの極端なケースがあるということです。Halkin は、$m=0$ の場合を finite horizon free end-point problem、$m=n$ で終端状態が固定される場合を finite horizon clamped end-point problem と呼んでいます。

有限期間で終端状態が自由な場合、Halkin は最大原理の条件のうち終端条件が

\[ \lambda(T)=0 \]

Important

All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me.

--- title: Be With You format: html: code-tools: true --- > 생각에 생각이 많아져 > > 엉켜버린 실타래 같지만 > > 너를 바라보는 이 순간의 느낌은 > > 너무나 행복한 걸 > > [*Be With You* - fromis_9](https://fromis9.fandom.com/wiki/Be_With_You) # No-Ponzi Game 条件（NPG）と横断性条件（TVC）の違い ## Starting Point: Household's Budget Constraint A representative household faces the following budget constraint: $$ b_{t+1} = (1+r_{t})b_{t} + y_{t} - c_{t} $$ This expression states that next period’s bond holdings are equal to the current assets carried forward with interest, plus current income, minus current consumption. By iterating this constraint forward indefinitely: $$ b_{t} = \sum^{\infty}_{j =0}\Biggl( \prod^{j-1}_{i = 0} \frac{1}{1+r_{t+i}} \Biggr) (y_{t+j} - c_{t+j}) + \lim_{j \rightarrow \infty} \Biggl(\prod^{j-1}_{i=0} \frac{1}{1+r_{t+i}} \Biggr) b_{t+j} $$ ## No-Ponzi Game (NPG) Condition A household cannot roll over its debt indefinitely by borrowing to finance both principal and interest. This restriction is necessary for the intertemporal budget constraint to hold and for the model to remain internally consistent. It rules out explosive debt paths that would imply an unbounded present value of liabilities. Accordingly, it functions as a feasibility condition of the model: $$ \lim_{j \rightarrow \infty} \Biggl(\prod^{j-1}_{i=0} \frac{1}{1+r_{t+i}} \Biggr) b_{t+j} = 0 $$ ## Transversality Condition (TVC) On the other hand, the condition derived from the household’s optimization problem is: $$ \lim_{t \rightarrow \infty} \beta^t \lambda_{t} b_{t+1} = 0 $$ This is the transversality condition, which ensures that the household does not find it optimal to accumulate unbounded debt or assets over the infinite horizon. It rules out explosive paths that would violate optimality, and it guarantees that the chosen consumption-saving plan satisfies the necessary boundary condition of the dynamic optimization problem. ## Mathematical relationship between two conditions From the first-order condition: $$ \beta^t \lambda_{t} = \beta^{t+1} \lambda_{t+1} (1+r_{t})= \beta^{t+1} u'(c_{t+1}) (1+r_{t}) $$ Substituting this expression into the transversality condition yields $$ \lim_{t \rightarrow \infty} \beta^{t+1} u'(c_{t+1}) (1+r_{t}) b_{t+1}= 0 $$ # なぜ transversality condition が必要なのか **有限期間問題と無限期間問題の違い** 横断性条件は単なる「終端条件」ではなく、無限遠に残された状態変数・資産・資本・資源の価値をどう扱うかという、無限期間問題に固有の条件として位置づけられます。 [Halkin(1974)](https://www.jstor.org/stable/1911976) は古典的な最適制御問題では終端時点があらかじめ与えられている場合でもそうでない場合でも、常に実数であると説明しています。これに対して経済学では終端時点が $+\infty$ になる無限期間問題を扱うため、数学的困難が生じると述べています。有限期間問題では時間区間を $I = [0, T]$ と置き、状態変数を $x(t)$, 制御変数を $u(t)$ とします。データとして閉集合 $Q \subset \mathbb R^r$, 関数 $(f, L): \mathbb R^n \times Q \times I \rightarrow \mathbb R^n \times \mathbb R$ を考えます。ここで $f$ は状態方程式、$L$ は瞬時利得とします。有限期間の最適制御問題は抽象的には $$ \max_{u(\cdot)} \int^T_0 L(x(t), u(t), t) dt $$ subject to $$ \dot(x)(t)=f(x(t), u(t), t), \quad u(t) \in Q, \quad x(0)=x_0 $$ という形で理解できます。重要な点は有限期間問題には終端条件の違いによって、少なくとも2つの極端なケースがあるということです。Halkin は、$m=0$ の場合を finite horizon free end-point problem、$m=n$ で終端状態が固定される場合を finite horizon clamped end-point problem と呼んでいます。有限期間で終端状態が自由な場合、Halkin は最大原理の条件のうち終端条件が $$ \lambda(T)=0 $$ ::: {.callout-important} All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me. :::