Financial exclusion and inflation costs

Abstract

This paper constructs two models of financial exclusion to assess the welfare costs of inflation. In the first, inflation costs are measured within a classical endowment economy. The second includes a production sector and costly credit. Both models are calibrated to account for inflation costs in a high-inflation economy (developing country) and in a low-inflation economy (developed economy). In an endowment economy, when inflation is reduced from 1.5% to zero in a developed economy, the welfare costs for agents with (without) financial access are 0.38% (0.43%) consumption equivalent variation (CEV). In a model with costly credit, the welfare costs for agents with (without) financial access are 0.87% (1.3%) CEV. For developing countries, when inflation is reduced from 3.2% to zero, the welfare costs for agents with (without) financial access are 0.72% (2.56%) in an endowment economy. In the costly-credit model, the welfare costs for agents with (without) financial access are 0.3% (3.1%) CEV. The main finding is that there is a substantial asymmetry in welfare costs between individuals with and without access to financial services, especially in developing countries.

Appendices

Steady states

1.1 Pure exchange economy

The system of equations that characterize the steady state in the pure exchange economy is:

$$\begin{aligned} c^B_{1} + c^B_{2} + \pi m^B = w^B +\frac{1-\beta }{\beta }b^B \\ c^B_{1} + \pi m^B = w^B + \frac{1-\beta }{\beta }b^B \\ c^B_{1} = m^B \\ c^B_{1} = m^B \\ \lambda ^B(c^B_{1} + c^B_{2}) + \lambda ^Bc^B_{1} = w^B + w^B \\ \lambda ^Bb^B+\lambda ^Bb^B = 0 \\ c_{2}^B = \frac{1}{\beta }(1+\pi )c^B_1 \end{aligned}$$

where $r=1/\beta -1$ in the steady state.

1.2 Economy with production and costly credit

In the steady state $r = 1/\beta - \delta$ and $n^g(s^B) = 1 - \int _0^{s^B}{\gamma (j)dj}$. Therefore we can write aggregate capital and wages as a function of $s^B$ such that

$$\begin{aligned} K(s^B) = \left( \frac{\alpha \beta }{1-\delta \beta }\right) ^{\frac{1}{1-\alpha }}n^g(s^B) \quad \text {and}\quad w(s^B) = (1-\alpha )\left( \frac{K(s^B)}{n^g(s^B)}\right) ^\alpha . \end{aligned}$$

The system of equations that characterize the steady state is:

$$\begin{aligned} c^B + w(s^B)\int _0^{s^B}{\gamma (j)dj} + \pi m^B = w(s^B) +\frac{1-\beta }{\beta }k^B \\ c^B + \pi m^B = w(s^B) + \frac{1-\beta }{\beta }k^B \\ c^B(1-s^B) = m^B \\ c^B = m^B \\ \lambda ^B(c^B + \delta k^B) + \lambda ^B(c^B+\delta k^B) = K(s^B)^\alpha n^g(s^B)^(1-\alpha ) \\ \lambda ^Bk^B+\lambda ^Bk^B = K(s^B) \\ w(s^B)\gamma (s^B) = [(1+\pi )/\beta - 1]c^B. \end{aligned}$$

Proof of Lemma 1

This proof is similar to the one found in Dotsey and Ireland (1996). Let $\beta ^t\lambda _t$ be the Lagrange multiplier on the budget constraint and $\beta ^t\phi _t$ be the Lagrange multiplier on the cash-in-advance constraint. Then, the first order conditions from the type-B agent lead to:

$$\begin{aligned} c^{B0}_t(j)&= u^{-1}_c(\lambda _t + \phi _t) \end{aligned}$$

(B1)

$$\begin{aligned} c^{B1}_t(j)&= u^{-1}_c(\lambda _t) \end{aligned}$$

(B2)

$$\begin{aligned} \xi _t^B(j)&= {\left\{ \begin{array}{ll} 1&{}\quad \text { if } u(c^{B1}_t) - \lambda _t[c^{B1}_t + q_t(j)] \ge u(c^{B0}_t) - (\lambda _t+\phi _t)c^{B0}_t \\ 0&{}\quad \text { otherwise}. \end{array}\right. } \end{aligned}$$

(B3)

Moreover, profit maximization for the firm in the intermediary sector leads to the following supply choice:

$$\begin{aligned} \xi _t^s(j) = {\left\{ \begin{array}{ll} 1&{}\quad \text { if } q_t(j) \ge w_t\gamma (j) \\ 0&{}\quad \text { otherwise}. \end{array}\right. } \end{aligned}$$

(B4)

note that Eq. (B4), together with the zero profit condition for the credit service market and the equilibrium for this market implies that

$$\begin{aligned} q_t(j) = w_t\gamma (j), \end{aligned}$$

(B5)

for all j demanded by type-B individuals, i.e., $\xi _t^B(j) = 1$.

Let $s_t^{B}\in {\mathcal {J}}$ be the good for which type-B individuals are indifferent between buying with credit or money, such that the inequality in Eq. (B3) holds with equality. Substituting Eqs. (B1), (B2) and (B4) into Eq. (B3) at equality yields:

$$\begin{aligned} \gamma (s^B_t) = \frac{u[u^{-1}_c(\lambda _t)] - \lambda _tu^{-1}_c(\lambda _t) - u[u^{-1}_c(\lambda _t + \phi _t)] + (\lambda _t+\phi _t)u^{-1}_c(\lambda _t + \phi _t)}{\lambda _tw_t},\nonumber \\ \end{aligned}$$

(B6)

which defines $s_t^B$.