2. The model

There are two periods, today and tomorrow, labeled as subscripts $t=0,1$ . At date 1, there are two states, $s=\alpha ,\beta $ with positive probabilities $0<\pi _{\alpha }<1$ and $\pi _{\beta }=1-\pi _{\alpha },$ respectively. There are two consumers, labeled as superscripts $i\in I=\left\{ B,S\right\} $ where B and S represent the asset buyer and the seller, respectively. Consumer i’s consumption allocation is $x^{i}=(x_{0}^{i},x_{\alpha }^{i},x_{\beta }^{i}) \in X=\mathbb{R}_{++}^{3}$ corresponding to prices $p=( p_{0},p_{\alpha },p_{\beta}) \gg 0.$ His endowment is $e^{i}=( e_{0}^{i},e_{\alpha}^{i},e_{\beta }^{i}) \in X$ where $e_{\alpha }^{i}=e_{\beta}^{i}=e_{1}^{i}.$ Consumer i’s utility functions, $v^{i}(x^{i}),$ are twice differentiable, strictly increasing, strictly concave, having the closure of indifference curves contained in $\mathbb{R}_{++}^{2}$ and satisfying the von Neumann–Morgenstern expected utility hypothesis. Consumer i’s preferences are

\begin{equation*}u^{i}(x^{i})=u^{i}\left( x_{0}^{i},x_{\alpha }^{i},x_{\beta }^{i}\right)=\pi _{\alpha }v^{i}\left( x_{0}^{i^{}},x_{\alpha }^{i}\right) +\pi _{\beta}v^{i}\left( x_{0}^{i},x_{\beta }^{i}\right) .\end{equation*}

Throughout this paper, we assume that there is an incentive to trade:

\begin{equation*}\frac{\partial v^{B}\left( e_{0}^{B},e_{1}^{B}\right) }{\partial x_{0}}/\frac{\partial v^{B}\left( e_{0}^{B},e_{1}^{B}\right) }{\partial x_{1}}<\frac{\partial v^{S}\left( e_{0}^{S},e_{1}^{S}\right) }{\partial x_{0}}/\frac{\partial v^{S}\left( e_{0}^{S},e_{1}^{S}\right) }{\partial x_{1}}.\end{equation*}

This condition implies that the initial endowment is not Pareto efficient. Since money can be considered a risky asset with sunspots, a lender and a borrower can be translated into an asset buyer and an asset seller.

In a monetary market, there is only one financial instrument (money) where $m^{i}$ denotes consumer i’s money holdings. Equilibrium in the monetary market is defined as follows: There are some positive spot prices $(p_{0},p_{\alpha },p_{\beta })\gg 0$ and associated money holdings m such that each household solves the following maximization problem:

\begin{equation*}\max \ \ u^{i}\!\left(x_{0}^{i},x_{\alpha }^{i},x_{\beta }^{i}\right)\end{equation*}

\begin{equation*}\text{subject to } \, \left\{\begin{array}{c}p_{0}x_{0}^{i}+m^{i}\leq p_{0}e_{0}^{i} \\ \ \ \ \ \ p_{\alpha }x_{\alpha }^{i}\leq p_{\alpha }e_{1}^{i}+m^{i} \ \ \ \ \ \ \ \ \\ \ \ \ \ p_{\beta }x_{\beta }^{i}\leq p_{\beta }e_{1}^{i}+m^{i}\ \ \ \ \ \ \ \end{array}\right.\end{equation*}

\begin{equation*}\text{and} \, x^{i}\in X,\end{equation*}

and each market clears:

\begin{equation*}\sum\nolimits_{i}x_{0}^{i}=\sum\nolimits_{i}e_{0}^{i},\text{ }\sum\nolimits_{i}x_{\alpha }^{i}=\sum\nolimits_{i}x_{\beta}^{i}=\sum\nolimits_{i}e_{1}^{i}\end{equation*}

\begin{equation*}\text{ and } \,\sum\nolimits_{i}m^{i}=0.\end{equation*}

We now introduce inflation-indexed bonds with a relative transaction cost $\theta $ . With the existence of transaction costs, the selling price of the bond is not the same as the purchasing price. The cost is incurred at date 1, implying that the intermediaries exchange the transaction fees with the first period goods. Therefore, the market clearing condition for the first period is dependent on the amount of the indexed bonds traded in the market. We label a monetary market with indexed bonds as IB. The equilibrium in IB is defined as

Given prices $(p_{0},p_{\alpha },p_{\beta },p)$ , $(m^{i},n^{i},x_{0}^{i},x_{\alpha }^{i},x_{\beta }^{i})_{h=1}^{H}$ solve the following maximization problem (P-IB)

\begin{equation*}\max u^{i}\!\left(x_{0}^{i},x_{\alpha }^{i},x_{\beta }^{i}\right)\end{equation*}

subject to

(1) \begin{equation}p_{0}x_{0}^{i}+m^{i}+p\left( 1+\theta \right) \max (n^{i},0)+p\min(n^{i},0)\leq p_{0}e_{0}^{i},\end{equation}

(2) \begin{equation}p_{\alpha }x_{\alpha }^{i}\leq p_{\alpha }e_{1}^{i}+p_{\alpha }n^{i}+m^{i},\end{equation}

(3) \begin{equation}p_{\beta }x_{\beta }^{i}\leq p_{\beta }e_{1}^{i}+p_{\beta }n^{i}+m^{i}.\end{equation}

Market clear conditions are

\begin{equation*}\sum\nolimits_{i}m^{i}=0,\sum\nolimits_{i}n^{i}=0,\end{equation*}

\begin{equation*}\sum\nolimits_{i}x_{0}^{i}+\frac{p}{p_{0}}\theta \sum\nolimits_{i}\max(n^{i},0)=\sum\nolimits_{i}e_{0}^{i},\end{equation*}

\begin{equation*}\text{and }\sum\nolimits_{i}x_{\alpha }^{i}=\sum\nolimits_{i}x_{\beta}^{i}=\sum\nolimits_{i}e_{1}^{i},\end{equation*}

where $n^{i}$ represents the amount of indexed-bond holdings of consumer i. $p\left( 1+\theta \right) $ and p in equation (1) represent the purchasing and selling prices of the indexed bonds, respectively. The nominal payoffs of the indexed bonds are $p_{\alpha }$ and $p_{\beta }$ in state $\alpha $ and $\beta ,$ respectively, as shown in equations (2)–(3).

The budget constraint set with indexed bonds in equations (1)–(3) is still convex even with the transaction cost, which is necessary for proof of the existence of single-valued demand functions. In the next section, this paper investigates the existence and regularity of the equilibria in a economy with indexed bonds.

3. Equilibrium

The degree of inflation volatility in this paper is defined as a relative standard deviation of inflation:

(4) \begin{equation}\sigma ^{R}=\frac{\sigma (\widetilde{p}_{1}/p_{0})}{E(\widetilde{p}_{1}/p_{0})},\end{equation}

where $\widetilde{p}_{1}$ is the random variable whose values are $p_{\alpha }$ with probabilities $\pi _{\alpha }$ and $p_{\beta }$ with probability $\pi _{\beta }$ , respectively. $\sigma (\widetilde{p}_{1}/p_{0})$ and $E(\widetilde{p}_{1}/p_{0})$ are the standard deviation and the expected value of the level of inflation, respectively.Footnote 4 Kang (Reference Kang2015) indicates that generically in endowment, for any given $\sigma ^{R}\geq 0$ , there exists a finite number of equilibria in a monetary economy with sunspots. This section shows that with the addition of the indexed bonds, generically in endowments, a finite number of equilibria do still exist for any given $\sigma ^{R}$ .

As fixing the degree of inflation volatility level ( $\sigma ^{R}$ ) as constant, the real returns of money can be derived. The following equation shows that there is a one-to-one relationship between $\sigma ^{R}$ and price ratio $\mathcal{P}$ ( $=p_{\beta }/p_{\alpha })$ :

\begin{equation*}\sigma ^{R}=\frac{\sigma (\widetilde{p}_{1}/p_{0})}{E(\widetilde{p}_{1}/p_{0})}=\frac{\sqrt{\pi _{\alpha }\pi _{\beta }}\left\vert 1-\mathcal{P}\right\vert }{\pi _{\alpha }+\pi _{\beta }\mathcal{P}}.\end{equation*}

For a given price ratio $\mathcal{P}$ , the ratio of the two states’ real returns of money is determined by the following relationship:

\begin{equation*}\mathcal{P}=\frac{p_{\beta }}{p_{\alpha }}=\frac{1/p_{\alpha }}{1/p_{\beta }}\end{equation*}

where $(\frac{1}{p_{\alpha }},\frac{1}{p_{\beta}})$ is the real return of money in states $\alpha $ and $\beta $ .

Given price ratio $\mathcal{P}$ , it is possible to interpret nominal security (money) as a security with a real return $\left( R_{\alpha},R_{\beta }\right) $ satisfying that $\mathcal{P}=R_{\alpha }/R_{\beta }$ . Then, the equivalent economy with two types of real securities can be defined.Footnote 5 The equivalent maximization problem based on two types of real securities is defined as (E-IB):

(5) \begin{equation}\max \text{ }u^{i}\!\left(e_{0}^{i}+z_{0}^{i},e_{1}^{i}+n^{i}+R_{\alpha}b^{i},e_{1}^{i}+n^{i}+R_{\beta }b^{i}\right)\end{equation}

(6) \begin{equation}\text{subject to } \, q_{0}z_{0}^{i}+qb^{i}+p\left( 1+\theta \right) \max(n^{i},0)+p\min (n^{i},0)\leq 0\end{equation}

\begin{equation*}\text{where } \,\frac{R_{\beta }}{R_{\alpha }}=\frac{1}{\mathcal{P}}\text{,}\end{equation*}

\begin{equation*}e_{0}^{i}+z_{0}^{i}=x_{0}^{i},\ \ e_{1}^{i}+n^{i}+R_{\alpha }b^{i}=x_{\alpha}^{i},\ e_{1}^{i}+n^{i}+R_{\beta }b^{i}=\ x_{\beta }^{i},\text{ }b^{i}=m^{i}/q,\text{ }\end{equation*}

and $b^{i}$ represents the amount of risky-asset holdings of consumer i. $z_{0}^{i}$ represents the excess demand for period-0 goods. q and $q_{0}$ represent the price of the real security and the price of the period-0 good, respectively.

The following lemma indicates that the maximization problem with risk assets ( $b^{i}$ ) is equivalent to that with money ( $m^{i}$ ):

Lemma 1 Given $\mathcal{P}=p_{\beta }/p_{\alpha }$ , P-IB is equivalent to E-IB where

\begin{eqnarray*}x_{0}^{i}\! &\!\!=e_{0}^{i}+z_{0}^{i},& \ \ \ x_{\alpha}^{i}=e_{1}^{i}+n^{i}+R_{\alpha }b^{i}, \ \ \ x_{\beta}^{i}=e_{1}^{i}+n^{i}+R_{\beta }b^{i}, \\ p_{0} &=q_{0}\left( =1\right),& \ \ \ p_{\alpha }=\frac{q}{R_{\alpha }},\ \ \ \text{and} \ \ \ p_{\beta }=\frac{q}{R_{\beta }},\end{eqnarray*}

Proof.

1. (i) (P-IB $\Rightarrow $ E-IB) P-IB is defined with the variables $\left(p_{0},p_{\alpha },p_{\beta },p,\mathcal{P}\right) $ while the variables are $(q_{0},q,R_{\alpha },R_{\beta },p,\mathcal{P})$ in E-IB. Defining $p_{0}$ and $m^{i}$ as

   \begin{equation*}p_{0}\equiv q_{0}\; \;\text{and } \, m^{i}\equiv qb^{i},\end{equation*}
   it is clear that the budget constraint in equation (6) in E-IB can be derived from that in P-IB. Next, defining $p_{\alpha }$ and $p_{\beta}$ as
   \begin{equation*}p_{\alpha }\equiv \frac{q}{R_{\alpha }},\ \ p_{\beta }\equiv \frac{q}{R_{\beta }},\end{equation*}
   $x_{\alpha }^{i}$ and $x_{\beta }^{i}$ can be expressed as
   \begin{equation*}x_{\alpha }^{i}\leq e_{1}^{i}+n^{i}+R_{\alpha }b^{i}\text{ and }x_{\beta}^{i}\leq e_{1}^{i}+n^{i}+R_{\beta }b^{i}.\end{equation*}
   Because utility functions are strictly increasing, $\leq $ can be replaced with $=$ , by the equation $x_{0}^{i}=e_{0}^{i}+z_{0}^{i}$ , the utility function (equation (5)) in E-IB is the same as the one in P-IB.

2. (ii) (E-IB $\Rightarrow $ P-IB) defining $q_{0}$ and $b^{i}$ as

   \begin{equation*}q_{0}\equiv p_{0}\ \ \text{and } \,b^{i}\equiv \frac{m^{i}}{q},\end{equation*}
   the period-0 budget constraint in equation (1) in P-IB is equivalent to that in equation (6) of E-IB. Defining $x_{\alpha}^{i}$ and $x_{\beta }^{i}$ as
   \begin{equation*}x_{\alpha }^{i}\equiv e_{1}^{i}+n^{i}+R_{\alpha }b^{i} \text{ and }x_{\beta}^{i}\equiv e_{1}^{i}+n^{i}+R_{\beta }b^{i},\end{equation*}
   $x_{\alpha }^{i}$ and $x_{\beta }^{i}$ can be expressed as
   \begin{equation*}\begin{array}{c}p_{\alpha }x_{\alpha }^{i}\leq p_{\alpha }e_{1}^{i}+p_{\alpha }n^{i}+m^{i}\\p_{\beta }x_{\beta }^{i}\leq p_{\beta }e_{1}^{i}+p_{\beta }n^{i}+m^{i},\end{array}\end{equation*}
   where
   \begin{equation*}R_{\alpha }=\frac{q}{p_{\alpha }},\ \ R_{\beta }=\frac{q}{p_{\beta }},\end{equation*}
   which are the same as equations (2) and (3), respectively. Because $x_{0}^{i}\equiv e_{0}^{i}+z_{0}^{i}$ , the utility function in P-IB is equivalent to that in E-IB.

With competitive equilibrium, the mean of the real return $(R_{\alpha},R_{\beta })$ does not affect the equilibrium allocations since the value of q is adjusted according to the value of the mean. For the convenience of proofs and computation, we assume that the mean value of the real return is fixed as one, that is, $\pi _{\alpha }R_{\alpha }+\pi _{\beta }R_{\beta }=1$ . So far, the analyses are based on the equivalent maximization problem (E-IB).

Proof. In the equivalent maximization problem, the budget sets are convex and the utility function is strictly concave for all $(q_{0},R_{\alpha },R_{\beta},q,\mathcal{P},p)\gg 0$ . Therefore, the excess demand is single valued. (The proof can be done by defining a compact subset of the budget set.)

For the proof of a regular and determinate economy, we need to define the aggregate excess demand and check if it satisfies sufficient conditions for the existence of regular economies. The market aggregate demand function is defined as

\begin{equation*}Z(Q) =\sum\nolimits_{i}\left(\begin{array}{c}z_{0}^{i}(Q) +\theta \frac{p}{q_{0}}\max \left( n^{i}\left(Q\right) ,0\right) \\b^{i}\left( Q\right) \\n^{i}\left( Q\right)\end{array}\right) ,\text{ }\end{equation*}

\begin{equation*}\text{where }Q=\left( q_{0},q,p\right) .\end{equation*}

This market aggregate demand function is not the same as a conventional one because of the additional term $\theta \frac{p}{p_{0}}\max (n^{i}(Q),0) $ which is the amount of first-period goods the intermediaries charge as transaction fees.

We can show that $Z\left( Q\right) $ satisfies the five properties:

1. (i) continuous.

2. (ii) homogeneous of degree zero.

3. (iii) (Walras’ law) $Q\cdot Z(Q)=0.$

4. (iv) There is an $s>0$ such that $Z(Q)>(-s,-s,-s)$ for all P.

5. (v) If $Q^{n}\rightarrow Q,$ where $Q\neq 0$ and $p=0$ , then $\sum\nolimits_{i}n^{i}(Q)\rightarrow \infty $ .

Therefore, the economy is regular and determinate.

The existence of equilibria does not necessarily imply that either (both) an indexed bond market or (and) a monetary market are active. Thus, this paper proves that the monetary market is always active even with the introduction of indexed bonds if the transaction cost is not zero as shown in the following proposition:

Proof. (by contradiction) Let’s assume that $m^{i}=0$ for $i\in \left\{S,B\right\} $ , labeling the asset buyer $(m^{i}>0)$ as $i=B$ and the asset seller $(m^{i}<0)$ as $i=S$ . Assuming that $n^{B}=n^{S}=0$ , the monetary market should be active since the initial endowment is not Pareto efficient. Next, we need to check the case where $n^{B}\neq 0$ and $n^{S}\neq 0$ . For this case, we can compute each individual’s value of the risky-asset (money). The value is the ratio between the marginal utility of the risk asset and that of the period-0 good. Let us assume that the risky asset’s return $\left( R_{\alpha },R_{\beta }\right) $ satisfies that $\pi _{\alpha}R_{\alpha }+\pi _{\beta }R_{\beta }=1$ . Then, the buyer’s value of the risky asset is computed as

(7) \begin{eqnarray}q_{R}^{B} &=&\frac{\pi _{\alpha }\frac{\partial v^{B}\left(x_{0}^{B},\;e_{1}^{B}+R_{\alpha }b^{B}\right) }{\partial b^{B}}+\pi _{\beta}R_{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;e_{1}^{B}+R_{\beta}b^{B}\right) }{\partial b^{B}}}{\pi _{\alpha }\frac{\partial v^{B}\left(x_{0}^{B},\;x_{\alpha }^{B}\right) }{\partial x_{0}}+\pi _{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\beta }^{B}\right) }{\partial x_{0}}}\notag \\&=&\frac{\pi _{\alpha }R_{\alpha }\frac{\partial v^{B}\left(x_{0}^{B},\;x_{\alpha }^{B}\right) }{\partial x_{1}}+\pi _{\beta }R_{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\beta }^{B}\right) }{\partial x_{1}}}{\pi _{\alpha }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\alpha }^{B}\right)}{\partial x_{0}}+\pi _{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\beta}^{B}\right) }{\partial x_{0}}}\text{ }\end{eqnarray}

Since $m^{B}=0$ (i.e., $b^{B}=0)$ , we have $x_{\alpha }^{B}=x_{\beta }^{B}$ . Also, it is true that $\pi _{\alpha }R_{\alpha }+\pi _{\beta }R_{\beta}=\pi _{\alpha }+\pi _{\beta }$ because we assume $\pi _{\alpha }R_{\alpha}+\pi _{\beta }R_{\beta }=1$ . Therefore, equation (7) can be expressed as

(8) \begin{equation}q_{R}^{B}=\frac{\pi _{\alpha }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\alpha}^{B}\right) }{\partial x_{1}}+\pi _{\beta }\frac{\partial v^{B}\left(x_{0}^{B},\;x_{\beta }^{B}\right) }{\partial x_{1}}}{\pi _{\alpha }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\alpha }^{B}\right) }{\partial x_{0}}+\pi_{\beta }\frac{\partial v^{B}\left( x_{0}^{i},\;x_{\beta }^{i}\right) }{\partial x_{0}}}.\end{equation}

The right-hand side of equation (8) is the same as the purchasing price of the indexed bond. Thus, we have

(9) \begin{equation}q_{R}^{B}=p(1+\theta ).\end{equation}

In the same way, we can show that the asset seller’s value of a risky-asset is the same as the selling price of the indexed bond:

(10) \begin{equation}q_{R}^{S}=p.\end{equation}

Because we know $q_{R}^{B}>q_{R}^{S}$ from equations (9)–(10), there is a price $q\in \left( q_{R}^{S},q_{R}^{B}\right) $ in which the two consumers have an incentive to trade with money. This contradicts that $m^{i}=0$ for $i=S,B$ .

Proposition 1 shows that the market for money is still operating at any level of inflation volatility if $\theta >0$ . However, without a transaction cost, that is, $\theta =0$ , the existence of inflation volatility makes the market for money completely shut down, as shown in Goenka and Préchac (Reference Goenka2006).

4. Indexed bonds and inflation volatility

This section analyzes how transaction cost and inflation volatility, respectively, affect the indexed bond market. With the transaction cost, the budget set is not smoothly defined so it is difficult to directly analyze how the market is affected by the transaction cost. Therefore, this section first investigates the economy without the indexed bond. Even if inflation-indexed bonds are not issued in a monetary market, the lender’s and borrower’s values of those securities can be computed at equilibrium. If the gap between the buyer’s and seller’s values for indexed bonds in the pure monetary economy is greater than the transaction cost, the consumer would have an incentive to trade on indexed bonds when the bond is introduced. If the gap is not large enough, the indexed bond market will not be active. In this case, the equilibrium allocation in a pure monetary market would be identical to that with indexed bonds.

In a market with indexed bonds, let us define $p_{F}^{i}$ as the value of a risk-free asset for consumer $i\in \left\{ B,S\right\} $ :

(11) \begin{equation}p_{F}^{i}=\frac{\pi _{\alpha }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\alpha}^{i}\right) }{\partial x_{1}}+\pi _{\beta }\frac{\partial v^{i}\left(x_{0}^{i},\;x_{\beta }^{i}\right) }{\partial x_{1}}}{\pi _{\alpha }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\alpha }^{i}\right) }{\partial x_{0}}+\pi_{\beta }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\beta }^{i}\right) }{\partial x_{0}}},\end{equation}

which is the ratio between the marginal utility of risk-free assets and that of the period-0 good. The following lemma shows that there is an arbitrage between two values, $p_{F}^{B}$ and $p_{F}^{S},$ where “B” and “S” represent the asset buyer and the asset seller, respectively.

Lemma 3 If $\sigma ^{R}>0$ in a monetary market without the indexed bond, the asset buyer’s value of the risk-free asset is greater than the asset seller’s, that is,Footnote ⁷

\begin{equation*}p_{F}^{S}<p_{F}^{B}.\end{equation*}

Proof. $q^{+}$ represents the equilibrium price of the risky asset in a market with the indexed bond:

(12) \begin{equation}q^{+}=\frac{\pi _{\alpha }R_{\alpha }\frac{\partial v^{i}\left(x_{0}^{i},\;x_{\alpha }^{i}\right) }{\partial x_{1}}+\pi _{\beta }R_{\beta }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\beta }^{i}\right) }{\partial x_{1}}}{\pi _{\alpha }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\alpha }^{i}\right)}{\partial x_{0}}+\pi _{\beta }\frac{\partial v^{i}\left( x_{0}^{i},\;x_{\beta}^{i}\right) }{\partial x_{0}}},\;\; \text{where}\ \ \ \ \left(\begin{array}{c}\pi _{\alpha }R_{\alpha }+\pi _{\beta }R_{\beta }=1 \\\mathcal{P}=R_{\alpha }/R_{\beta }\end{array}\right)\end{equation}

Let us label the asset buyer $(b^{i}>0)$ as $i=B$ and the asset seller $(b^{i}<0)$ as $i=S$ . Since the initial endowment allocation is not Pareto efficient, they will trade and therefore $b^{B}>0$ and $b^{S}<0$ .

First, we want to show that $p_{F}^{B}$ is higher than $q^{+}$ . From equations (11) and (12), $p_{F}^{B}/q^{+}$ is

(13) \begin{equation}\frac{p_{F}^{B}}{q^{+}}=\frac{\pi _{\alpha }\frac{\partial v^{B}\left(x_{0}^{B},\;x_{\alpha }^{B}\right) }{\partial x_{1}}+\pi _{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\beta }^{B}\right) }{\partial x_{1}}}{\pi_{\alpha }R_{\alpha }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\alpha}^{B}\right) }{\partial x_{1}}+\pi _{\beta }R_{\beta }\frac{\partial v^{B}\left( x_{0}^{B},\;x_{\beta }^{B}\right) }{\partial x_{1}}}.\end{equation}

Since the state $\beta $ is inflationary (i.e., $\mathcal{P}>1)$ , we have $R_{\alpha }>R_{\beta }$ and $x_{\alpha }^{B}>x_{\beta }^{B}$ . Because $v^{i}$ is strictly concave and $x_{\alpha }^{B}>x_{\beta }^{B}$ , we have

(14) \begin{equation}\frac{\partial v^{B}\left( x_{0}^{B},x_{\alpha }^{B}\right) }{\partial x_{1}}<\frac{\partial v^{B}\left( x_{0}^{B},x_{\beta }^{B}\right) }{\partial x_{1}}.\text{ }\end{equation}

Since $\pi _{\alpha }R_{\alpha }+\pi _{\beta }R_{\beta }=1$ , $\pi _{\alpha}+\pi _{\beta }=1$ and $R_{\alpha }>R_{\beta }$ , we have $\pi _{\alpha}R_{\alpha }>\pi _{\alpha }$ , $\pi _{\beta }R_{\beta }<\pi _{\beta }$ . Thus, from equations (13)–(14) we have:

(15) \begin{equation}\frac{p_{F}^{B}}{q^{+}}>1.\end{equation}

In the same way, we can prove the following:

(16) \begin{equation}\frac{p_{F}^{S}}{q^{+}}<1\text{.}\end{equation}

From the two inequalities, we know that $p_{F}^{S}<p_{F}^{B}$ .

The arbitrage in the values of the risk-free securities between two consumers provides the incentive for them to accept the real securities as the financial instrument. However, the existence of the arbitrage does not necessarily imply that the indexed bond market is active. The magnitude of the arbitrage should be large enough, compared to the transaction cost, for the indexed bonds to be traded in the financial market. The following propositions show that the indexed bond markets can be active or inactive depending on the level of inflation volatility and the level of the transaction cost.

Proposition 2 If $\sigma ^{R}>0$ , there exists $\overline{\theta }>0$ such that the indexed bond market is active (inactive) if $\theta <\overline{\theta }$ $\left( \theta \geq \overline{\theta }\right) $ where

(17) \begin{equation}\overline{\theta }=\frac{p_{F}^{B}-p_{F}^{S}}{p_{F}^{S}}.\end{equation}

Proof.

1. (i) For $n^{i}\neq 0$ where $\theta <\overline{\theta }$ : (Proof by contradiction) Let us assume that $n^{i}=0$ for $i=S,B$ . Then, the equilibrium allocations in the market with indexed bonds are the same as those in a pure monetary market. If $\theta <\overline{\theta }$ , there exists $p>0$ such that $p_{F}^{S}<p<p\left( 1+\theta \right)<p_{F}^{B} $ . (This can be shown as follows: Let $p=p_{F}^{S}+\varepsilon $ and $\theta =\left( p_{F}^{B}-p_{F}^{S}-\delta \right) /p_{F}^{S}>0$ . Then, for all $\delta \in \left( 0,p_{F}^{B}-p_{F}^{S}\right) $ , there exists $\varepsilon , $ which satisfies the inequality $p_{F}^{S}\leq p<p\left(1+\theta \right) \leq p_{F}^{B}$ ). This implies that the two consumers have an incentive to trade with the indexed bonds. This contradicts that $n^{i}=0$ .

2. (ii) For $n^{i}=0$ where $\theta \geq \overline{\theta }$ : (Proof by contradiction) Let us assume that $n^{i}\neq 0$ for $i=S,B$ . Assuming that $\theta \geq \overline{\theta }$ , there does not exist $p>0$ such that $p_{F}^{S}<p<p\left( 1+\theta \right) <p_{F}^{B}$ . This contradicts that $n^{i}\neq 0$ for $i=S,B$ .

The next question is how inflation volatility affects trading on the indexed bond. In the model, the decision about whether financial entrepreneurs should enter the market (or whether the government should issue indexed bonds) depends on the value of $\overline{\theta }$ . For a smaller value of $\overline{\theta }$ , there would be a smaller chance for financial entrepreneurs to survive in the market. The following proposition shows that $\overline{\theta }$ increases with the increased volatility of inflation:

Proof. See Appendix for the proof.

Proposition 3 is the extended version of Lemma 3 indicating that $\overline{\theta }$ is strictly positive where $\sigma ^{R}>0$ and equals zero where $\sigma ^{R}=0$ . The proof for the nonseparable utility functions remains open. Proposition 3 implies that as the economy has higher inflation volatility, financial entrepreneurs have a better chance of making a profit and, consequently, stronger incentive to enter the indexed bond market.

The proof in Proposition 3 assumes that the utility function is concave in $x_{1}$ regardless of the change in $x_{0}$ with increased volatility. With this assumption, this paper shows that as the market is exposed to higher inflation volatility (i.e., as $\mathcal{P}$ increases), the gap between the buyer’s and seller’s values of indexed bonds is increasing. With additively separable utility functions, the change of $x_{1}$ with increased volatility does not affect the concavity of the utility function, so we can easily show that the expected utility is concave in $\mathcal{P}$ . However, with nonseparable utility functions, the utility function is not necessarily concave in $x_{1}$ with the change of $\mathcal{P}$ .Footnote 8 Thus, this paper rules out the nonseparable utility-function cases in Proposition 3.

5. An example

This section presents a numerical example to help explain the main results in this paper. Assume that there are two consumers, an asset buyer (B) and asset seller (S) with endowments of (10,0) and (0,10), respectively. Their expected utility functions are the same as $\log (x_{0})+\log (x_{1})$ . State probabilities are $\pi _{\alpha }=\pi _{\beta }=0.5$ . Assuming that the inflation level $\mathcal{P}=2$ , the corresponding price-level (inflation) volatility is $\sigma ^{R}=33.3\%$ by equation (4). In this case, it is computed that $\overline{\theta }=23\%$ by equation (17). Assuming that the transaction cost is 10% $\left( \theta=0.1\right) $ in the market with indexed bond, the equilibrium outcomes are summarized in Table 1.

Table 1. Equilibrium outcomes in an economy without (and with) indexed bonds

The price levels in the monetary market are higher than those in the market with the indexed bonds. Higher price levels imply a higher value of money in period 0. As explained in Section 4, the higher value of money is attributed to the asset buyer’s higher demand for money. Consequently, the value of money in the economy with the indexed bonds is not as high as that in a pure monetary market. The change in money value after the introduction of indexed bonds causes the asset seller’s utility to decrease from $3.26$ to $3.18$ . (See Table 1.)

Figure 1 represents the equilibrium allocations in the space of excess demand. The figure should be three dimensional including the excess demand of $x_{0}$ , but we can imagine the original three-dimensional figure is projected onto a two-dimensional figure. The buyer’s allocation is located in the northeast quadrant, while the seller’s allocation is in the southwest quadrant. The allocations are symmetric to (0,0) by market clearing conditions. “+” and “o” represent the equilibrium allocations in pure monetary and indexed-bond markets, respectively. From the figure, we know that the equilibrium allocations move closer to the 45 degree line in the market with indexed bonds although both markets have the same volatility level.

Figure 1. Equilibrium allocations in the space of excess demand.

Figure 2 shows how the two consumers’ utility changes in transaction costs. The asset seller’s utility is increasing in transaction cost $\theta $ if $\theta >10\%$ . Higher transaction costs make the demand for money increase by the substitution effects. Therefore, the price of money goes up and, consequently, the asset buyer would have higher financial income from selling money (risky asset) at a higher price. If the transaction cost is higher than 23%, the indexed bond market would be inactive. In that case, the equilibrium allocations in the market with indexed bonds $\ $ are identical to those in a pure monetary economy.

Figure 2. Consumers’ utility changes in transaction costs.

6. Welfare analysis

The indexed bond market gives consumers more trading choices than a pure monetary market. Therefore, it can be expected that equilibrium with indexed bonds is Pareto superior to that without the bond. However, it has been shown that financial innovations to incomplete markets do not necessarily induce the economy to be Pareto improving (Elul (Reference Elul1995) and Cass and Citanna (Reference Cass1998)). Financial innovations are known to affect market prices under a general equilibrium setting. These price changes affect agents’ real wealth and, consequently, their utility values. Specifically, the introduction of indexed bonds causes the value of money to decrease due to a precautionary savings effect, which has a negative impact on asset borrowers’ wealth.

The example in Section 5 is the case where the introduction of indexed bonds make some consumers worse off. As the level of inflation volatility increases, the second consumer’s (i.e., borrower) utility value can increase, as shown in Section 5. The intuition is as follows. With a high level of inflation volatility, the real return of assets becomes small in an inflationary state. If the inflationary state is realized, the first consumer (i.e., saver) will end up getting a small amount of the good at date 1 and, consequently, his utility will be particularly low. Therefore, the first consumer’s demand for the asset will be higher to insure against the inflationary state, due to the precautionary savings effect. The high demand for savings results in a high value of money. Finally, the borrower obtains more income as the asset price becomes higher in a monetary market.

However, the introduction of new riskless securities could be a substitute for money and cause a decrease in money demand. Therefore, the value of money when indexed bond are available is smaller than that in a pure monetary market. A decrease in the money value negatively affects the borrowers’s utility level. In this example, even though the asset seller has more trading choices with the indexed bond, his utility level is actually lower than that in a pure monetary market. (See the Appendix for more details about the example.)

Although some of consumers may be negatively affected, it is not possible for all of them to be worse off with the introduction of indexed bonds according to the revealed preferences hypothesis. The following proposition shows this.

Proof.

1. (i) When the value of money, which is q in the equivalent maximization problem, is not the same in both markets, there must be at least one consumer who will benefit from the change in value. The consumer should be the asset buyer (seller) if the value decreases (increases). Since the consumer has more choices with the indexed bond, according to the revealed preference argument, the consumer must be better off.

2. (ii) See the example in Section 5.

The introduction of the indexed bonds can be considered a sunspot-stabilizing policy in the sense that it makes the equilibrium allocations less volatile.Footnote 9’Footnote 10 However, Proposition 4 indicates that the introduction of indexed bonds does not necessarily make all agents better off. This implies that the government can fail to gain consensus in adopting the financial innovation on indexed bonds. This section shows that in Proposition 5, if balanced lump-sum tax plans are allowed along with the financial innovation, consumer consensus on the policy can be achieved. Although there is no clear Pareto ranking between equilibrium allocation with and without the indexed bonds as shown in Proposition 4, this section shows that equilibrium with indexed bonds is Kaldor superior to that without the bond by considering balanced lump-sum tax plans. The space of the balanced lump-sum tax plan is defined as

\begin{equation*}T=\left\{ (\tau ^{1},\tau ^{2})\in\mathbb{R}^{2}|\tau ^{1}+\tau ^{2}=0\right\} .\end{equation*}

The tax-transfer plan $(\tau ^{1},\tau ^{2})$ is applied to the market with indexed bonds. Then, the budget constraints at date 0 are modified to be

(P-CM) $$s.t\;\left\{ {\matrix{ {{p_0}x_0^i + {m^i} + p\left( {1 + \theta } \right)\max ({n^i},0) + p\min ({n^i},0) \le {\mkern 1mu} {p_0}e_0^i - {p_0}{\tau ^i}} \cr {} \cr {{p_\alpha }x_\alpha ^i \le {p_\alpha }e_1^i + {p_\alpha }{n^i} + {m^i}} \cr {{p_\beta }x_\beta ^i \le {p_\beta }e_1^i + {p_\beta }{n^i} + {m^i}} \cr } } \right.$$

We define Kaldor improving as follows. The financial innovation on indexed bonds is Kaldor improving if there exist(s) $(\tau ^{1},\tau ^{2})$ $\in T$ such that the equilibrium allocations in the economy with the indexed bonds is Pareto superior to those without indexed bonds. The following proposition shows that the market with indexed bonds is superior to a pure monetary market based on the compensation test:

Proof. See the Appendix.

7. Concluding remarks

Government inflation-linked bonds have become available in a number of countries including the USA and UK, and they have attracted enormous research interest. However, most research papers are written based on partial-equilibrium standards. In partial equilibrium, it is trivial that the agents facing inflation uncertainty can increase welfare by trading in indexed bond markets. However, in general equilibrium, the introduction of indexed bonds can change other assets prices so it is not easy to answer whether (1) both lenders and borrowers are willing to trade in the indexed bonds market and (2) introduction of indexed bonds improves welfare. This paper answers both questions by incorporating transaction costs in trading on the indexed bonds with sunspots.Footnote 11

This paper shows that inflation-linked bonds can be more actively traded in an economy with higher inflation volatility. In the same way, we show that the uncertainty of a nominal interest rate can provide room for inflation-linked bonds. Using a two-period general equilibrium model, it may be possible to show that monetary authorities’ imperfect process of controlling the money supply can cause the economy to prefer indexed bonds. While inflation volatility is based on market psychology, volatility in the nominal interest rate is caused by an inconsistent monetary policy.

This paper shows that financial innovation in indexed bonds changes all price levels in a continuum of equilibria. Specifically, the introduction of indexed bonds endogenously changes the continuum of equilibrium price levels, which makes comparative statics difficult to conduct. Thus, this paper fixes the volatility level ( $\mathcal{P}$ ) so it can be interpreted as an “exogenous” variable. It is beyond the scope of this paper to show how the exogenous variable ( $\mathcal{P}$ ) is made endogenously, which is also nontrivial issue in the literature of incomplete markets.

Recently, a significant number of inflation-linked bonds have been issued by governments and private financial institutions. These bonds are taking up a more important position as widely accepted financial instruments. The findings of this study provide a better understanding of these bonds from a theoretical point of view.