21. Stockman's Cash-in-Advance Model (1981, Journal of Monetary Economics)

21.1 Set-up

Let's consider a pure endowment economy with only one consumption good where the endowment is the consumption good. The consumption good and one period bond (the only asset) are both traded at nominal prices with money. Specifically:

• Exogenous stochastic process for shocks \(\{s_t\}_{t=0}^{\infty}\) with \(s_t=(y_t,\omega_t)\) where:
  • \(y_t\) is the endowment at date \(t\);
  • \(\omega_t\) is the money supply growth rate (gross growth) at date \(t\): let \(\bar M_t\) be the money supply (by the central bank which doesn't appear in this model elsewhere) at date \(t\), then \(\omega_t\) is defined by \(\bar M_t=\omega_t\bar M_{t-1}\) \(\forall t\);
  • \(\{s_t\}_{t=0}^{\infty}\) follows a first-order Markov process, with transition function \(F(s';s)\) whose density is \(f(s';s)\).
• Timing: in each period \(t\), the following things happen in this order:
  • First, at the beginning of each period \(t\), the shock \(s_t=(y_t,\omega_t)\) is realized;
  • Second, asset market meets: to liquidate the transaction of last period; and to decide the bond holding and cash holding in this period;
  • Finally, the consumption good is purchased with cash holding in this period and consumption happens.
• Prices:
  • Price for the one period nominal bond whose payoff is 1 unit of money next period is \(Q_N(s_t,\bar M_t)\), which is a function of realized current state \((y_t,\omega_t)\) and money supply \(\bar M_t\);
  • Price for one unit of consumption good is \(P(s_t,\bar M_t)\).
• Money enters the economy through lump-sum transfers as the money supply grows in each period, i.e. the lump-sum transfer in period \(t\) is

$$ \bar M_t-\bar M_{t-1}=(\omega_t-1)\bar M_{t-1} $$

• Household's preferences are represented by the following utility function: \(\mathbb{E}\left[\sum_{t=0}^{\infty}\beta^t u(C_t)\right]\).

21.2 Strategies for solving the model

1. Set up the model; 2. Analyze the representative household's problem; 3. Define a recursive competitive equilibrium; 4. Solve for the equilibrium prices; 5. Discuss some examples.

21.3 Household's problem

21.3.1 Budget constraint (BC)

Notice that in this economy, liquidation of transactions for bonds happens immediately, but liquidations of transactions for consumption goods this period happens in the next period. For example, at date \(t-1\) the household bought \(C_{t-1}\) units of consumption good and sells \(y_{t-1}\) units of his endowment both at price \(P(s_{t-1},\bar M_{t-1})\) and bought \(N_{t-1}\) units of the one period nominal bond at price \(Q_N(s_{t-1},\bar M_{t-1})\), then the household pays \(Q_N(s_{t-1},\bar M_{t-1})N_{t-1}\) at date \(t-1\) and pays \(P(s_{t-1},\bar M_{t-1})(C_{t-1}-y_{t-1})\) and receives \(N_{t-1}\) at date \(t\). So, in each period \(t\), the household's budget constraint is

$$ M_t+Q_N(s_t,\bar M_t)N_t\le N_{t-1}+M_{t-1}+P(s_{t-1},\bar M_{t-1})(y_{t-1}-C_{t-1})+(\omega_t-1)\bar M_{t-1}\quad \forall t $$

where the LHS is the portfolio of cash and bond chosen in this period, and the RHS is bond payoff, cash carried over, net sales income and monetary transfer respectively. We can define the RHS to be a single variable: asset

$$ A_t\equiv N_{t-1}+M_{t-1}+P(s_{t-1},\bar M_{t-1})(y_{t-1}-C_{t-1})+(\omega_t-1)\bar M_{t-1}\quad \forall t $$

21.3.2 Cash-in-advance constraint (CIA)

We assume that this economy is a "back scratching" economy, which means that the household cannot consume his own endowment. They must exchange their endowments in order to consume the good. Such transactions must happen with money. So, this gives rise to the cash-in-advance constraint as

$$ P(s_t,\bar M_t)C_t\le M_t\quad \forall t $$

21.3.3 Conjectures

In a recursive competitive equilibrium that satisfies the Quantity Theory (footnote: Quantity Theory says that, in the long run, increase in money leads to one to one increase in prices), we can make the following conjectures about the prices:

• The nominal bond price can be written as

$$ Q_N(s_t,\bar M_t)=q(s_t) \tag{21.1} $$

where \(Q_N(s_t,\bar M_t)\) doesn't increase in \(\bar M_t\) because twice of money today also implies twice of money tomorrow, so they cancel out, and we only have a real term \(q(s_t)\) left in the bond nominal price. We can also define the nominal interest rate \(i(s_t)\) by \(Q_N(s_t,\bar M_t)=q(s_t)=\dfrac{1}{1+i(s_t)}\). Typically, it is reasonable to assume \(Q_N(s_t,\bar M_t)\le1\) and \(i(s_t)\ge0\), otherwise people would never buy such bond. In equilibrium, the nominal interest rate \(i(s_t)\) would be such that the net supply of this bond is zero (homogeneous households \(\Rightarrow\) zero buying and zero selling for each household in equilibrium).

• The consumption good price can be written as

$$ P(s_t,\bar M_t)=p(s_t)\bar M_t \tag{21.2} $$

where \(p(s_t)\) can be interpreted as the real component in the price that is determined by the current realized state, and \(\bar M_t\) is the nominal component in the price.

21.3.4 Household's Bellman equation

$$ V(A_t;s_t,\bar M_t)=\max_{C_t,M_t,N_t}\left\{u(C_t)+\beta\mathbb{E}_{s_{t+1}}\left[V(A_{t+1};s_{t+1},\bar M_{t+1})\,|\,s_t,\bar M_t\right]\right\} $$

or equivalently, we can expand the expectation:

$$ V(A_t;s_t,\bar M_t)=\max_{C_t,M_t,N_t}\left\{u(C_t)+\beta\int V(A_{t+1};s_{t+1},\bar M_{t+1})\,dF(s_{t+1};s_t)\right\} $$

$$ \text{s.t.}\quad M_t+Q_N(s_t,\bar M_t)N_t\le A_t\ \text{(BC)} \tag{21.3} $$

$$ P(s_t,\bar M_t)C_t\le M_t\ \text{(CIA)} \tag{21.4} $$

where \(A_{t+1}\equiv N_t+M_t+P(s_t,\bar M_t)(y_t-C_t)+(\omega_{t+1}-1)\bar M_t\) \(\forall t\).

21.3.5 Normalized Bellman equation

We can define the normalized variables \(a_t\equiv\dfrac{A_t}{\bar M_t}\), \(m_t\equiv\dfrac{M_t}{\bar M_t}\), \(n_t\equiv\dfrac{N_t}{\bar M_t}\). Then, by (21.1) and (21.2), we can divide by \(\bar M_t\) on both sides and rewrite the constraints (21.3) and (21.4) as \(m_t+q(s_t)n_t\le a_t\) and \(p(s_t)C_t\le m_t\). And then, we can rewrite the household's Bellman equation in a normalized form as

$$ V(a;s)=\max_{C,m,n}\left\{u(C)+\beta\mathbb{E}_{s'}\left[V(a';s')\,|\,s\right]\right\} \tag{21.5} $$

$$ \text{s.t.}\quad m+q(s)n\le a\ \text{(BC)} \tag{21.6} $$

$$ p(s)C\le m\ \text{(CIA)} \tag{21.7} $$

where

$$ a'\equiv\frac{1}{\omega'}\big[n+m+p(s)(y-C)+(\omega'-1)\big]\quad \forall t \tag{21.8} $$

21.4 Recursive competitive equilibrium

Again, recursive competitive equilibrium basically means that what the agents (households and firms) expect ex-ante will be realized ex-post as a result of the choices and actions of the agents. Here, the only type of agent making decisions is representative household, so their expectation should meet the outcome that clears all the markets (consumption good market, bond market and cash market) as a general equilibrium result.

• Market clearing condition for consumption good market is \(y=C\) (since the only supply and demand of consumption goods come from homogeneous households).
• Market clearing condition for bond market is \(n=0\) (since we can assume the only supply and demand of one period nominal bond come from homogeneous households; the case where government supplies and demands bond is trivial because government could realize its transaction in bond market by simply altering the money supply \(\bar M_t\) of this period and \(\bar M_{t+1}\) of the next period, so we should reserve the bond market purely for households).
• Market clearing condition for cash market is \(m=1\), i.e. \(M_t=\bar M_t\) (government supplies money, and household takes (demands) money, so in equilibrium supply should be equal to demand).

The representative household takes as given the prices \(p(s),q_N(s)\) and law of motion of states \(F(s';s)\). A recursive competitive equilibrium in this model consists of

• Prices: \(\{p(s),q_N(s)\}\)
• Functions: \(\{C(a,s),m(a,s),n(a,s)\}\)

such that market clearing condition holds for consumption, bond and cash. The recursiveness here comes in through \(a=1\). Consider the law of motion of asset (21.9):

$$ a'=\frac{1}{\omega'}\big[n+m+p(s)(y-C)+(\omega'-1)\big] \tag{21.9} $$

Plug in the three market clearing conditions to (21.9) to obtain

$$ a'=\frac{1}{\omega'}\big[0+1+p(s)\cdot0+(\omega'-1)\big]=1 \tag{21.10} $$

Note that (21.10) is true for any period, so the household would expect \(a=a'=1\) when making their decisions on \(\{C(a,s),m(a,s),n(a,s)\}\), and finally because of their choice of \(\{C(a,s),m(a,s),n(a,s)\}\), it would satisfy that \(a'=1\). So, we can rewrite the three market clearing conditions for the recursive competitive equilibrium:

$$ C(1,s)=y(s) \tag{21.11} $$

$$ n(1,s)=0 \tag{21.12} $$

$$ m(1,s)=1 \tag{21.13} $$

Then, the next goal is to characterize the recursive equilibrium prices \(\{p(s),q_N(s)\}\) that satisfies (21.11), (21.12), and (21.13).

21.5 Solve for the equilibrium prices

21.5.1 Form the Lagrangian of household's problem

We can write the equations (21.5), (21.6), (21.7) and (21.8) together as a Lagrangian:

$$ \mathcal{L}=u(C)+\beta\mathbb{E}_{s'}\left[V(a';s')\,|\,s\right]+\lambda(s)\big(a-m-q(s)n\big)+\mu(s)\big(m-p(s)C\big) \tag{21.14} $$

where

$$ a'=\frac{1}{\omega'}\big[n+m+p(s)(y-C)+(\omega'-1)\big]\quad \forall t \tag{21.15} $$

and \(\lambda(s)\) is the Lagrangian multiplier for BC and \(\mu(s)\) is the Lagrangian multiplier for CIA. Notice that the choice variables are \(C,m,n\).

21.5.2 Assumption: binding cash-in-advance constraint

We will focus on the economies where CIA constraint binds, which means that the Lagrangian multiplier \(\mu(s)\) for CIA is strictly positive. Then, household would not leave money unused in each period, because otherwise household could buy bond rather than hold cash without using it, so binding CIA also implies there is positive value for cash in hand, i.e. \(i(s_t)>0\). So,

$$ \text{CIA binds}\ \Leftrightarrow\ \mu(s)>0\ \Leftrightarrow\ i(s_t)>0\ \Leftrightarrow\ q(s)<1 $$

With this assumption, we have that \(p(s)C=m\), which, in equilibrium, is equivalent to

$$ p(s)y(s)=1\quad \text{for }\forall s \tag{21.16} $$

(21.16) implies that \(p(s)\) only depends on \(y(s)\), not \(\omega\). (21.16) also implies that

$$ \text{velocity}\equiv\frac{p(s)\bar M_t C}{\bar M_t}=p(s)C=p(s)y(s)=1 $$

where velocity is defined as total volume of transaction divided by total money supply. So, the velocity of money is always 1 in equilibrium with the assumption of binding CIA.

21.5.3 First-order conditions and envelop condition

There are three first-order conditions, which are for \(C,m\) and \(n\) respectively. And there is one envelop condition for \(a\). Note that for markets clearing in equilibrium, all the conditions should hold when evaluated at \(C=y\), \(m=1\) and \(n=0\).

• First-order condition for \(C\) is

$$ \begin{aligned} u'(C)+\beta\mathbb{E}_{s'}\left[\frac{-p(s)}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]+\mu(s)(-p(s))&=0\\ \Rightarrow\ u'(C)&=\beta\mathbb{E}_{s'}\left[\frac{p(s)}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]+\mu(s)\,p(s)\\ \Rightarrow\ u'(y)&=p(s)\left[\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]+\mu(s)\right] \end{aligned} \tag{21.17} $$

Plug in (21.19) below to (21.17) to get

$$ u'(y)=p(s)\lambda(s) \tag{21.18} $$

• First-order condition for \(m\) is

$$ \begin{aligned} \beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]-\lambda(s)+\mu(s)&=0\\ \Rightarrow\ \mu(s)&=\lambda(s)-\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right] \end{aligned} \tag{21.19} $$

Plug in (21.21) below to (21.19) to get

$$ \mu(s)=\lambda(s)\big(1-q(s)\big) \tag{21.20} $$

• First-order condition for \(n\) is

$$ \begin{aligned} \beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]-q(s)\lambda(s)&=0\\ \Rightarrow\ q(s)\lambda(s)&=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right] \end{aligned} \tag{21.21} $$

Plug in (21.25) below to (21.21) to get

$$ q(s)\lambda(s)=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}\lambda(s')\,\Big|\,s\right] \tag{21.22} $$

• Envelop condition: first, establish the law of motion of asset as

$$ A_{t+1}=\frac{A_t-M_t}{q(s_t)}+M_t+P(s_t,\bar M_t)(y_t-C_t)+(\omega_{t+1}-1)\bar M_t\quad \forall t $$

where the next period asset \(A_{t+1}\) is the bond payoff \(\frac{A_t-M_t}{q(s_t)}\) from this period's investment \(A_t-M_t\), plus the zero-interest-bearing cash \(M_t\) carried over, plus this period's net profit from consumption good trading \(P(s_t,\bar M_t)(y_t-C_t)\) that is liquidated next period, plus the cash transfer \((\omega_{t+1}-1)\bar M_t\). So,

$$ \begin{aligned} \Rightarrow\ \frac{A_{t+1}}{\bar M_{t+1}}&=\frac{1}{\bar M_{t+1}}\left[\frac{A_t-M_t}{q(s_t)}+M_t+P(s_t,\bar M_t)(y_t-C_t)+(\omega_{t+1}-1)\bar M_t\right]\\ \Rightarrow\ a_{t+1}&=\frac{1}{\omega_{t+1}}\left[\frac{a_t-m_t}{q(s_t)}+m_t+p(s_t)(y_t-C_t)+(\omega_{t+1}-1)\right]\\ \Rightarrow\ a'&=\frac{1}{\omega'}\left[\frac{a}{q(s)}+m\left(1-\frac{1}{q(s)}\right)+p(s)(y-C)+(\omega'-1)\right] \end{aligned} \tag{21.23} $$

Then, consider the Bellman equation \(V(a;s)=\max_{C,m,n}\{u(C)+\beta\mathbb{E}_{s'}[V(a';s')\,|\,s]\}\), replace \(a'\) in the Bellman equation with (21.23) and take derivative on both sides w.r.t. \(a\) to get the envelop condition

$$ V_a(a;s)=\beta\mathbb{E}_{s'}\left[\frac{1}{q(s)}\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right]=\frac{1}{q(s)}\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}V_{a'}(a';s')\,\Big|\,s\right] $$

Plug in (21.21) to obtain

$$ V_a(a;s)=\lambda(s) \tag{21.24} $$

and similarly

$$ V_{a'}(a';s')=\lambda(s') \tag{21.25} $$

So far, we have established that \(p(s)\) only depends on \(y(s)\), and \(q(s)\lambda(s)=\beta\mathbb{E}_{s'}\big[\frac{1}{\omega'}\lambda(s')\,|\,s\big]\).

21.6 Examples

21.6.1 Example 1: both income and money supply growth are constant

Suppose \(y(s)=\bar y\) and \(\omega=\bar\omega\). Then, by (21.16), \(p(s)=\frac{1}{\bar y}\) which is a constant. And since all states are the same, the Lagrangian multipliers \(\lambda(s)=\bar\lambda\) and \(\mu(s)=\bar\mu\) are constant. So, by (21.22),

$$ q(s)\bar\lambda=\beta\mathbb{E}_{s'}\left[\frac{1}{\bar\omega}\bar\lambda\,\Big|\,s\right]\ \Rightarrow\ q(s)=\frac{\beta}{\bar\omega} $$

is also a constant. Note that by the assumption of binding CIA, we imposed that \(q(s)<1\). Let \(\beta=\frac{1}{1+\rho}\) and \(\bar\omega=1+\pi\). Then, \(q(s)<1\) implies

$$ \frac{\frac{1}{1+\rho}}{1+\pi}<1\ \Rightarrow\ (1+\pi)(1+\rho)>1 \tag{21.26} $$

When \(\pi\) and \(\rho\) are not very big, (21.26) is equivalent to \(\pi+\rho>0\), which means that deflation rate (absolute value of \(\pi\) when \(\pi\) is negative) cannot be too big. This is very intuitive since if deflation is very severe, then household would want to hold cash (so cash becomes a store of value instead of transaction tool) and never buy consumption good, and thus the market cannot clear, which makes the model not suitable.

21.6.2 Example 2: income is constant, money supply growth is stochastic

Suppose income is constant, i.e. \(y(s)=\bar y\), and money supply growth \(\omega\) is stochastic. Then, by (21.16) again, we still have that

$$ p(s)=\frac{1}{\bar y} \tag{21.27} $$

which is a constant. And by (21.18) and (21.27), the Lagrangian multiplier

$$ \lambda(s)=u'(\bar y)\bar y=\bar\lambda \tag{21.28} $$

is also a constant. Plug (21.28) to (21.22) to obtain

$$ q(s)\bar\lambda=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}\bar\lambda\,\Big|\,s\right]\ \Rightarrow\ q(s)=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}\,\Big|\,s\right] \tag{21.29} $$

where (21.29) implies that the bond price \(q(s)\) and thus nominal interest rate \(i(s)\) only depend on expected money growth rate, and they depend on this period's realized money growth rate \(\omega\) only through the effect of \(\omega\) on next period's expected money growth rate \(\omega'\).

21.6.3 Example 3: both income and money supply growth are stochastic

Suppose both income and money supply growth are stochastic. Then, in this example we assume Constant Relative Risk Aversion (CRRA) utility

$$ u(C)=\frac{C^{1-\theta}-1}{1-\theta} $$

where \(\theta>0\); when \(\theta\to1\), \(u(C)=\log C\). Then, by (21.16), \(p(s)=\frac{1}{y(s)}\) which is stochastically pinned down by \(y(s)\). And by (21.18) and (21.16),

$$ \lambda(s)=\frac{u'(y)}{p(s)}=u'(y)\,y=y^{1-\theta} $$

and similarly \(\lambda(s')=(y')^{1-\theta}\). By (21.22), we have that

$$ q(s)\lambda(s)=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}\lambda(s')\,\Big|\,s\right]\ \Rightarrow\ q(s)=\beta\mathbb{E}_{s'}\left[\frac{1}{\omega'}\left(\frac{y'}{y}\right)^{1-\theta}\,\Big|\,s\right] \tag{21.30} $$

Note that (21.30) tells us current period price of bond depends on the expected growth rate of money supply in next period, and depends on the expected real income growth rate.

• When \(\theta>0\) and \(\theta\ne1\), there are two opposing forces in the real economy (income) part that drives \(q(s)\):
  • Higher real income today (i.e. higher \(y\)) \(\Rightarrow\) Lower price \(p(s)=\frac{1}{y(s)}\) today \(\Rightarrow\) Higher expected inflation \(\Rightarrow\) Higher \(i(s)\) \(\Rightarrow\) Lower \(q(s)\);
  • Higher real income today (i.e. higher \(y\)) \(\Rightarrow\) Lower expected real income growth \(\Rightarrow\) Lower expected real interest rate \(\Rightarrow\) Lower \(i(s)\) \(\Rightarrow\) Higher \(q(s)\).
  • Note that both stories end up having lower \(\left(\frac{y'}{y}\right)\), but the results are opposite, which implies that the first one corresponds to \(\theta<1\) and the second one corresponds to \(\theta>1\).
• When \(\theta=1\), the bond price does not depend on real income anymore, and it only depends on expected money supply growth rate.