Consumption

A representative agent of the economy chooses his consumption path as to maximize the following utility criterion:

U_t=

∞

X

s=t

β^s−tlog(C_s) (3.1)

To abstract from inessential dynamics, it is assumed that β, the subjective discount factor, equals the real interest rate factor, (1 + r)⁻¹. C denotes a Cobb-Douglas consumption index over tradable (C_T) and non-tradable (C_N) consumer goods

C_t = C_T,t^γ C_N,t^1−γ. (3.2) γ ∈ (0, 1) and (1 − γ) are the weights of tradable and non-tradable goods.⁹ Following Calvo and V´egh (1993), it is assumed that consumption has to be covered with money holdings from the previous period:

PT,tCT,t+ PN,tCN,t≤ Mt−1 (3.3)

9This formulation of the consumption basket implies a unity elasticity of substitution between tradables and non-tradables

_{d log}^CT CN



d log_pN

pT

|_{U = ¯U} = 1

 .

where M_t−1 denotes period t − 1 nominal money holdings and P_T,t and P_N,t the domestic currency prices of tradable and non-tradable final goods, re-spectively. The cash-in-advance constraint captures the function of money as a medium of exchange and can be thought to result from sequenced open-ing hours of goods and asset markets.¹⁰ The above cash-in-advance con-straint assumes that transactions in both the market for tradable and for non-tradable goods are conducted with domestic currency. An alternative formulation of the cash-in-advance constraint consists in requiring agents to hold either domestic or foreign currency for transacting, thus accounting for currency substitution (see Calvo and V´egh, 1994b). However, this implies that the amount of seignorage accruing to the foreign central bank, and thus the economy’s aggregate wealth, decreases in the inflation rate. In order to avoid these wealth effects, currency substitution is ruled out in this chapter’s model.¹¹ Note that the cash-in-advance constraint holds with equality when the nominal interest rate is positive: Agents do not hold money in excess of what is strictly necessary for consuming if they can earn interest on holding bonds instead. Since attention is restricted to equilibria with positive interest rates, the cash-in-advance constraint therefore implies that

PT,tCT,t+ PN,tCN,t = Mt−1. (3.4) The agent’s nominal period budget constraint is given by

P_tB_t+1^∗ = P_t(1 + r)B_t^∗− (M_t− M_t−1) + P_tC_t+ P_T,tY¯_T + p_N,t(z)y_N,t(z) + P_tg_t. (3.5) P_t, B_t+1^∗ , ¯Y_T and g_t denote the price level, that is, the consumption-based price index (which will be explicitly derived later), real holdings of foreign bonds at the beginning of period t + 1, the endowment of tradable goods and government net transfer to the private sector, respectively. y_N,t(z) denotes the output produced by the firm which is owned by the agent indexed by z, p_N,t(z) the price.¹² Equation (3.5), coupled with the transversality condition

10Clower (1967) is the pioneering work incorporating cash-in-advance constraints. De-tailed treatments of sequenced opening hours in a stochastic environment can be found in Sargent (1987:158ff). The notation for prices used in the cash-in-advance constraint, that is, explicitly denoting the price levels of tradable and of non-tradable goods is adopted for the sake of expositional clarity. It is particularly useful for analyzing (temporary) changes in the nominal exchange rate. No fundamental difference exists between this notation and the more common practice of formulating everything in terms of the relative price of tradable goods.

11For the same reason, imports are not modeled to be subject to a foreign-cash-in-advance constraint.

12Notice that pN,t(z)yN,t(z) is the sum of the nominal production cost and the profits of intermediate non-tradable goods production. The latter are given as

lim_{T →∞}(1 + r)^−T 

B_{t+T +1}^∗ +^M_P^{t+T +1}

t+T +1



= 0 and the assumption that agents’

initial asset holdings equal zero yields the intertemporal budget constraint as

∞ This chapter’s analysis will show that the price charged and quantity supplied by each monopolist equal the aggregate non-tradables’ price, P_N,t, and output of non-tradable final goods, Y_N,t. In anticipation on this result, p_N,t(z) and y_N,t(z) are therefore substituted with P_N,t and Y_N,t in what follows.

Substituting the cash-in-advance constraint into the period budget con-straint the latter can be reformulated to yield

C_t= −P_t−1 Substituting this expression for C into the intertemporal utility function and taking the derivative with respect to B^∗ gives rise to the following first-order condition for optimal consumption:

C_t P_t

P_t−1 = C_t+1P_t+1

P_t (3.7)

Equation (3.7) differs from the usual Euler equation as agents have to hold currency in the period prior to consumption. Defining the inflation rate be-tween periods t − 1 and t as π_t ≡ _P^Pt

t−1 − 1, π_t+1 as π_t+1 ≡ ^P_P^t+1

t − 1, the first-order condition can be reformulated to yield

∂U (C_t)/∂C_t

1 + π_t = ∂U (C_t+1)/∂C_t+1

1 + π_t+1 (3.8)

where ∂U (Ct)/∂Ct denotes the derivative of utility with respect to consump-tion, ∂U (C_t)/∂C_t= C_t⁻¹. The above equation states that the marginal utility of consumption per unit of its cost must be equal across periods. Since con-sumption must be covered by the preceding period’s money holdings, the cost of consumption includes the cost of holding money, that is, for money held between periods t − 1 and t the inflation rate π_t, and π_t+1 for money held

pN,t(z)yN,t+j(z) − Pt+jM CyN,t+j(z) where M C y_N,t+j(z) denotes the real production cost function.

between periods t and t + 1. The first order condition shows that optimal consumption is constant in equilibria with a constant inflation rate.¹³

The first-order condition and the intertemporal budget constraint allow to derive a closed-form solution for optimal consumption as

C_t= (1 + π_t)⁻¹ A detailed derivation of this expression for consumption can be found in ap-pendix 3.6.1.

First-order condition (3.7) determines the optimal path for total consump-tion. The division of total consumption between tradable and non-tradable goods is found by maximizing total consumption, given total expenditure, that is, by maximizing

C_t= C_T,t^γ C_N,t^1−γ subject to

P_N,t

P_t C_N,t+P_T,t

P_t C_T,t = Z_t

where Z_t equals total real expenditure on consumption. This yields¹⁴ C_T,t= γ 1

where Z_t^nom denotes total nominal expenditure, Z_t^nom = P_tZ_t. Total real ex-penditure, Z_t, equals total real consumption, C_t. Therefore, above demand equations can be expressed as

C_T,t= γ P_t P_T,tC_t

13Recall that the subjective discount factor and the real interest rate factor do not enter the first-order condition, as these were assumed to be equal.

14The derivation is as follows: In a first step, the Lagrange function L = C_{T ,t}^γ C_N,t^1−γ− λ(^P_P^N,t

t CN,t+^P_P^{T ,t}

t CT ,t− Zt) is maximized with respect to CT ,t and CN,t. The ratio of first-order conditions gives relative consumption of non-tradables as ^C_C^N,t

T ,t =

1−γ γ

PN,t

P_{T ,t}. Substituting CN and CT with the values implied by the total expenditure function yields equations (3.9) and (3.10).

and

CN,t= (1 − γ) P_t P_N,tCt.

In a last step, the (consumption-based) price index P is derived. Follow-ing Obstfeld and Rogoff (1996:227), it is defined as the minimum nominal expenditure Z_t^nom required to acquire one unit of the consumption bundle.¹⁵ Formally, this can be expressed as

P_t≡ {Z_t^nom|C(Z_t^nom) = 1} .

Demand equations (3.9) and (3.10) maximize consumption, given spending Z^nom. The minimum expenditure to acquire one unit of C is thus found by substituting these demand functions into the definition of the consumption basket, equation (3.2), setting the resulting expression equal to one.

 γ 1

P_T,tZ_t^nom

γ

(1 − γ) 1

P_N,tZ_t^nom

1−γ

= 1 Solving the above equation for Z_t^nom yields the price index as

P_t= P_T,t γ

γ P_N,t 1 − γ

(1−γ)

. (3.11)