22. Variant of Cash-in-Advance Model

There could be multiple ways to alter the set-up in classic Cash-in-Advance model in section 21, such as changing the timing of the model. In this section, we are focusing on the model that relaxes the CIA constraint in (21.7) but introduces other costs of transactions induced by not using cash.

22.1 Set-up

• Four types of agents in this model: household, bank, firms, and government.
• One good produced by capital and labor.
• Three means of payments:
  • cash: happens most frequently (highest number of transactions); used for small (small volume means each single transaction of this type involves small amount of payment) volume transaction. Here "volume" of a transaction is the same as the "size" of a transaction, which means the amount of payment for one particular transaction.
  • card: happens less frequently (medium number of transactions); used for medium volume transaction.
  • electronic transactions: happens least frequently (least number of transactions); used for large volume transaction.
• Household:
  • has preferences represented by utility function \(u(c)\) where \(c\) is the consumption level that cannot be changed for a short period of time, i.e. suppose household makes decision once in each period with length \(\tau\); \(\rho\) is the discounting parameter for utility; if the consumption level is chosen as \(c\), then the present value of utility in that short period is

$$ \int_0^{\tau}e^{-\rho t}u(c)\,dt=-\frac{1}{\rho}e^{-\rho t}u(c)\Big|_0^{\tau}=\frac{1}{\rho}\big(1-e^{-\rho\tau}\big)u(c) $$

- has fixed distribution \(G(z)\) for the relative size of purchases (here the distribution is fixed for the relative size, not the absolute size, so we are allowing the absolute volume to adjust for different consumption levels):
    - here the distribution doesn't embed any stochastic component. Instead, the distribution simply means the fraction of transactions in each size range, which is deterministic;
    - \(G(z)\) has density function \(g(z)\): \(z\) is the relative volume (or size) of a transaction; \(g(z)\) assigns positive density to having a transaction of volume \(z\); again, this density is just the fraction of purchases at each volume, not a probability density, and the problem is deterministic, not stochastic; we define density only because this model is set up in a continuous fashion.
- we can define sum relative volume of purchases of relative volume (or size) \(z\) or less, which is \(\tilde G(z)\), the added up relative volume of all transactions whose single relative volume is less or equal to \(z\), i.e.

$$ \tilde G(z)=\int_0^z \zeta\cdot g(\zeta)\,d\zeta $$

- Let \(\nu=\lim_{z\to\infty}\tilde G(z)\) be the total relative volume of all possible purchases.
- Let \(\Omega(z)\) denote the fraction of total relative volume contributed by transactions whose individual relative volume is equal to or less than \(z\), then

$$ \Omega(z)=\frac{\tilde G(z)}{\nu} $$

- Let \(n^h\) be the total number of transactions for the household:
    - we assume that alternations in consumption level \(c\) are carried out only by adjusting the absolute size of transactions (intensive margin), not by adjusting the total number of transactions (extensive margin);
    - so, the total number of transactions \(n^h\) is not varying with \(c\), and the relative volume distribution of those \(n^h\) transactions is fixed.

22.2 Household's problem

22.2.1 Decisions to be made

The household will be choosing the means of payment for different purchases. In particular, the household will choose two thresholds to cut the relative volume axis into three segments such that each transaction in segment take different means of payment. The characteristics of each payment method (time cost is the time spent on carrying out the transaction by a particular payment method, in terms of time, not forgone wages unless multiplied with wage rate; liquidity requirement is defined as the fraction that needs to be held liquid as cash so that this part of asset cannot earn interest):

• Cash: No time cost \(k^c=0\); highest liquidity requirement \(\theta^c=1\).
• Card: medium time cost (\(0=k^c
• Electronic payment: highest time cost \(k^e>k^d>k^c=0\); lowest liquidity requirement \(0<\theta^e<\theta^d<\theta^c=1\).

Note that the time cost is only associated with the number of transactions for each payment method, not the individual volume of each transaction. So,

• for transactions with high frequency and low individual size, household would use cash;
• for transactions with low frequency and high individual size, household would use electronic payment;
• for transactions in between, household would use card.

Therefore, instead of making decisions of payment method for each individual transaction, household determines two threshold of relative volume \(\gamma\) and \(\delta\) (\(0<\gamma<\delta\)) such that

• household use cash if the transaction's relative size is below \(\gamma\);
• household use card if the transaction's relative size is between \(\gamma\) and \(\delta\);
• household use electronic payment if the transaction's relative size is over \(\delta\).

22.2.2 Liquidity requirement and total time cost

Define the total liquidity requirement by (since \(\theta^c=1\))

$$ L^h(\gamma,\delta)=\Omega(\gamma)\cdot\theta^c+[\Omega(\delta)-\Omega(\gamma)]\cdot\theta^d+[1-\Omega(\delta)]\cdot\theta^e=\Omega(\gamma)+[\Omega(\delta)-\Omega(\gamma)]\cdot\theta^d+[1-\Omega(\delta)]\cdot\theta^e \tag{22.1} $$

• By definition in (22.1), \(L^h(\gamma,\delta)\) is the fraction of total absolute transaction volume to be made by liquid asset (i.e. cash) that cannot earn interest.
• So, \(P\,c\,L^h(\gamma,\delta)\) is the total payment by cash, and \(P\,c\,(1-L^h(\gamma,\delta))\) is total payment by interest-bearing asset, where \(P\) is the nominal price of consumption good.

Define the total time cost by (since \(k^c=0\))

$$ T^h(\gamma,\delta)=G(\gamma)\cdot k^c+[G(\delta)-G(\gamma)]\cdot k^d+[1-G(\delta)]\cdot k^e=[G(\delta)-G(\gamma)]\cdot k^d+[1-G(\delta)]\cdot k^e \tag{22.2} $$

• By definition in (22.2), \(T^h(\gamma,\delta)\) is the time cost of one transaction.
• So, \(n^h T^h(\gamma,\delta)\) is the total time spending on \(n^h\) purchases.

22.2.3 Household Bellman equation

$$ V(A_0;P_0)=\max_{M_0,\gamma,\delta,\tau,c}\left\{\frac{1}{\rho}\big(1-e^{-\rho\tau}\big)u(c)+e^{-\rho\tau}V(A';P_0)\right\} \tag{22.3} $$

$$ \text{s.t.}\quad c\,L^h(\gamma,\delta)\cdot\frac{1}{\pi}\big(e^{\pi\tau}-1\big)\le M_0 \tag{22.4} $$

where \(M_0\) is the cash-in-advance money chosen for this period with length \(\tau\), and \(\pi\) is the (constant) rate of increase in money supply and thus rate of increase in consumption good price (by Quantity Theorem in equilibrium).

The law of motion of asset is (the three lines are: cash left over, saving, net accumulated wage respectively):

$$ A'=\underbrace{\left[M_0-c\,L^h(\gamma,\delta)\,P_0\cdot\frac{1}{\pi}\big(e^{\pi\tau}-1\big)\right]}_{\text{cash left over}} \tag{22.5} $$

$$ +\underbrace{e^{i\tau}\big(A_0-M_0-P_0 w\nu\phi\big)}_{\text{saving}} \tag{22.6} $$

$$ +\underbrace{\frac{e^{\pi\tau}-e^{i\tau}}{\pi-i}P_0\big[w(1-n^h T^h(\gamma,\delta))-f-c(1-L^h(\gamma,\delta))\big]}_{\text{net accumulated wage}} \tag{22.7} $$

where \(i\) is the nominal interest rate, \(w\) is the real wage and \(f\) is the real lump-sum tax (levied by government) in terms of real consumption good, \(\phi\) is the time spent on a trip to bank to withdraw cash. So,

• line (22.5) is the cash not used up in purchasing, which earns zero nominal interest;
• line (22.6), i.e. asset \(A_0\) at the beginning minus cash \(M_0\) to be hold and minus bank trip cost \(P_0 w\nu\phi\), is the saving that is untouched throughout the period of length \(\tau\);
• line (22.7) is the total nominal value of the accumulated labor income net of real tax and consumption real expenditure through non-cash payment:
  • note that the household is using \((1-n^h T^h(\gamma,\delta))\) fraction of time to work in each infinitesimally small time period to pay for lump-sum tax and consumption;
  • so, \(w(1-n^h T^h(\gamma,\delta))-f-c(1-L^h(\gamma,\delta))\) is the real net labor income to be accumulated after paying for tax and consumption (not-by-cash part) in the infinitesimally small time period, which is denoted by \(acc\):

$$ acc=w(1-n^h T^h(\gamma,\delta))-f-c(1-L^h(\gamma,\delta)) $$

- this part of asset is accumulated in nominal term that both has nominal interest rate return and increasing price.

Now we have reached the Bellman equation (22.3), the CIA constraint (22.4), and the law of motion of asset (22.5), (22.6) and (22.7), then the rest of the analysis of this model can be carried out in a similar way as the model discussed in section 21 except that everything is deterministic in this model.