Micro Hedging vs. Macro Stability: A Unified Theory of Currency Invoicing Under Exchange Rate Risk

3.4.1 Analytical Content

The threshold is derived endogenously from the firm's expected profit maximisation under CES preferences and monopolistic competition. It balances two countervailing forces: the convexity of the profit function in the exchange rate, which always favours PCP (Proposition 1), and the concavity introduced by the cost function when α is large. To find the balance point, we combine Jensen's inequality with the log-moment generating function of a log-normal. Appendix A.2 shows that the qualitative result holds well outside log-normality. Once distributions are described by their cumulants, including skewed and fat-tailed ones, the threshold barely moves, and α and the LCP incentive stay monotonically related across the board. The value 0.5, in other words, is an artefact of the log-normal assumption, not of the economics. The threshold likewise nests in the general arbitrage condition of Proposition 6: set every covariance term to zero, and what survives is ½ Var(s)(2α − 1) ≥ 0 for LCP stability, so the partial-equilibrium result is a true limiting case. The robustness analysis and the cumulant-based derivation of the zone of indifference are in Appendix A.2–A.4.

3.4.2 Practical Limitations

Three frictions, in practice, carve out a zone of indifference around the threshold—a range over which firms hold their invoicing strategy fixed. Menu costs come first: switching the invoicing currency of a contract incurs administrative, legal, and relational costs, and only an expected profit gain that exceeds them justifies the change (Ball & Romer, 1990; Mankiw, 1985). Contractual inertia comes second, because trade contracts typically lock in the currency for the contract's life, sometimes years; the threshold binds only when the contract is renewed. Buyer power and industry convention come third. In sectors with concentrated buyers or settled pricing norms—commodities quoted in dollars being the textbook example, the currency is often dictated by the buyer or by convention, regardless of the exporter's cost structure.

Empirically, the upshot is that a test of Proposition 2 should anticipate a smooth, non-linear increase in LCP adoption as α rises above the threshold rather than an abrupt jump at 0.5. Threshold regressions, or specifications with flexible functional forms, suit the task—a point Prediction P1 (Section 5) takes up. A fuller treatment of the zone of indifference, one that incorporates switching costs, contract length, and bargaining power, is left for future work.

4.1.1 Household Preferences

The representative household in each country consumes a final consumption basket comprising final goods from both countries. For the domestic household:

C t = c H t n c F t 1 − n n n ( 1 − n ) 1 − n , 0 < n < 1

(22)

Here C H t and C F t are baskets of domestic and foreign final goods, respectively. The elasticity of substitution between domestic and foreign aggregates is unity. The parameter n represents the effective size of the domestic economy (Corsetti & Pesenti, 2001). Within each category ( H or F ), the household chooses from a continuum of differentiated goods:

C H t = [ ∫ 0 1 c H t θ − 1 θ ( i ) d i ] θ θ − 1

(23)

C F t = [ ∫ 0 1 c F t θ − 1 θ ( i ) d i ] θ θ − 1

(24)

The resulting demand functions and price indices are derived from expenditure minimisation.

The domestic household allocates consumption between domestic and foreign goods varieties. The optimal demands for each variety are given by:

P H t C H t = n P t C t

(25)

P F t C F t = ( 1 − n ) P t C t

(26)

These lead to the consumer price index:

P t = P H t n P F t 1 − n

(27)

Within each variety, the household chooses among differentiated goods. The optimal demand for good j is:

c H t d ( j ) = [ p H t ( j ) p H t ] − θ C H t

(28)

c F t d ( j ) = [ p F t ( j ) p F t ] − θ C F t

(29)

The corresponding price indices are the price indices and :

P H t = [ ∫ 0 1 p H t 1 − θ ( i ) d i ] 1 1 − θ

(30)

P F t = [ ∫ 0 1 p F t 1 − θ ( i ) d i ] 1 1 − θ

(31)

Intertemporal Program: The household maximises expected utility:

U ( C t , H t ) = log ( C t ) − η H t

(32)

Money is introduced via a cash-in-advance constraint:

P t C t ≤ M t

(33)

P t C t + M t + ∫ v ( s t + 1 ) B ( s t + 1 ) d s t + 1 ≤ P t w t H t + M t − 1 + T t + B t + D t + ∫ π ( j ) d j

(34)

The household's intertemporal program is formulated in the Bellman equation form:

V ( B t ( s t ) , M t − 1 ) = max { U ( C t , H t ) + β ∫ V ( B t + 1 ( s t + 1 ) , M t ) f ( s t + 1 , s t ) d s t + 1 }

(35)

In each period, the household has income from labour P t w t H t , government transfers T t , nominal cash holdings M t − 1 , and financial securities B t resulting from choices in the previous period. As the owner of domestic firms in both sectors, the household receives dividends at the end of the period, denoted respectively as D I t for the intermediate goods sector and ∫ π ( j ) d j for the final goods sector. It allocates its resources to consumption expenditures, demands for monetary balances, and purchases of financial securities. The financial markets are complete ² . For each state of nature s t + 1 , there exists a contingent claim B ( s t + 1 ) which, when purchased in period t, yields the household one unit of domestic currency if the state s t + 1 materialises. The price associated with the state s t + 1 is υ ( s t + 1 ) . The function f ( s t + 1 , s t ) determines the transition from one state s t to another state s t + 1 .

The first-order conditions are as follows:

C t − 1 = P t ( λ t + μ t )

(36)

η = w t P t λ t

(37)

λ t = β E t [ 1 P t + 1 C t + 1 ]

(38)

∫ υ ( s t + 1 ) d s t + 1 = β E t [ λ t + 1 λ t ]

(39)

With complete markets, we obtain the international risk-sharing condition:

λ t * = S t λ t

(40)

The central bank increases the money supply through direct transfers to households. The government's budget constraint in each country is:

M t = M t − 1 + T t and M t * = M t − 1 * + T t *

(41)

Following Devereux and Engel (2001), we assume the money supply follows a random walk process:

E t [ M t M t + 1 ] = 1

(42)

With complete markets, logarithmic preferences, and the random walk assumption, the equilibrium exchange rate is:

S t = M t M t *

(43)

This is a deliberately stylised setup—the nominal exchange rate tracks relative money supplies exactly, with no overshooting. The payoff is a clean reading of how exchange rate movements feed into firms’ pricing, and the algebra stays manageable. Technical details are provided in Appendix B.2.

Perfectly competitive firms produce homogeneous intermediate goods using labour:

Q t = A I t H t

(44)

Here Q t = Q H t + Q H t * represents total production, sold in both domestic and foreign markets; A I t is the level of technology in the sector; and H t is the labour input, supplied by domestic households (labour is immobile across countries).

The law of one price holds for intermediate goods. Profit maximisation yields:

w t = v t P t A I t

(45)

w t * = v t * P t * A I t *

(46)

Monopolistically competitive firms produce differentiated final goods using domestic and foreign intermediates:

Y H t ( j ) = A t Q F t α ( j ) Q H t 1 − α ( j ) , 0 < α < 1

(47)

Cost minimisation yields the unit cost function:

C T t min ( j ) = κ v t 1 − α ( S t v t * ) α A t Y t ( j )

(48)

Firms in the final goods sector sell output to consumers in both countries. The firm's production, Y t ( j ) , is thus distributed between the domestic and foreign markets, depending on the quantities demanded by domestic households, c H t d ( j ) , and foreign households, c H t * d ( j ) :

c H t d ( j ) = n ( p H t ( j ) P H t ) − θ P t C t P H t

(49)

c H t * ( j ) = ( 1 − n ) ( p H t * ( j ) P H t * ) − θ P t * C t * P H t *

(50)

We analyse the scenario where all domestic firms coordinate on the same invoicing strategy. Consider a domestic firm i in isolation. We use the domestic household's marginal utility λ because domestic firm owners value foreign-market profits in domestic currency, converted at the exchange rate through the international risk-sharing condition λ * = S λ .

Symmetric PCP Equilibrium:

If all domestic firms adopt PCP, firm i sets price p H t E ( i ) to maximise expected profit. The first-order condition yields:

p H t E ( i ) = θ θ − 1 E t − 1 [ S t ( θ − 1 ) P H t * ( θ − 1 ) P t * C t * λ t * M C t ] E t − 1 [ S t ( θ − 1 ) P H t * ( θ − 1 ) P t * C t * λ t * ]

(56)

In symmetric equilibrium, p H t E ( i ) = P H t E = S t P H t * , which simplifies to:

p H t E ( i ) = θ θ − 1 E t − 1 [ P t * C t * λ t * M C t ] E t − 1 [ P t * C t * λ t * ]

(57)

The optimal price applies a constant markup to expected marginal cost, weighted by foreign wealth indicators. The coefficient ϕ t − 1 is derived from the profit maximisation process. Starting from the first-order condition and substituting the optimal price back into the profit function, we obtain after simplification: ϕ t − 1 = β ( 1 − n ) θ λ t − 1 , where n represents the share of foreign market sales and θ the elasticity of substitution. Substituting into the profit function gives maximum expected profit:

π H t E = ϕ t − 1 E t − 1 [ P t * C t * λ t * ]

(58)

Symmetric LCP Equilibrium:

Under symmetric LCP, the optimal foreign-currency price is:

p H t I ( i ) = θ θ − 1 E t − 1 [ P H t * ( θ − 1 ) P t * C t * λ t * M C t ] E t − 1 [ P H t * ( θ − 1 ) P t * C t * λ t * S t ]

(59)

In the symmetric equilibrium, where domestic firms choose to price their goods in the buyer's currency, the pricing strategy becomes p H t I ( i ) = P H t I = P H t * , simplifying to:

p H t I ( i ) = P H t I = θ θ − 1 E t − 1 [ P t * C t * λ t * S t − 1 M C t ] E t − 1 [ P t * C t * λ t * S t − 1 ]

(60)

The maximum expected profit is therefore:

π H t I = ϕ t − 1 E t − 1 [ P t * C t * λ t * ]

(61)

Equations (58) and (61) deliver an indifference result that should be read with care. The two expected profits match not because the invoicing currency does not matter, but because a lone deviating firm, in a symmetric equilibrium, is too small to shift aggregate price indices. With every other firm pricing in one currency, that currency alone composes the aggregate index. Since P = SP* in the symmetric PCP equilibrium, a firm that moves to LCP faces a price-index comparison it would not face outside symmetry. Indifference is thus a knife-edge property of complete coordination—not a sign that the currency choice is inconsequential. The interesting case is its collapse: contemplate a unilateral deviation and the deviation-incentive conditions of Propositions 3 and 4 emerge, and it is these that determine equilibrium stability.

It is the unilateral deviation that destroys the indifference. An equilibrium holds only if no firm gains by abandoning the strategy the others have settled on.

Deviation from Symmetric PCP:

If all other firms use PCP, we examine whether firm i profits by switching to LCP. The deviating firm's profit is:

π H t E d e v ( i ) = E t − 1 [ β λ t λ t − 1 { [ S t P H t E d e v ( i ) − M C t ] ( 1 − n ) ( p H t E d e v ( i ) S t − 1 P H t E ) − θ × S t P t * C t * P H t E ] }

(62)

The optimal deviation price satisfies:

p H t E d e v ( i ) = θ θ − 1 E t − 1 [ P t * C t * λ t * S t − θ M C t ] E t − 1 [ P t * C t * λ t * S t 1 − θ ]

(63)

The expected profit is:

π H t E d e v ( i ) = ϕ t − 1 [ E t − 1 ( P t * C t * λ t * ) E t − 1 ( P t * C t * λ t * M C t ) ] 1 − θ [ E t − 1 ( P t * C t * λ t * S t − θ M C t ) ] 1 − θ [ E t − 1 ( P t * C t * λ t * S t 1 − θ ) ] 1 − θ

(64)

Proposition 3

(Deviation from PCP): A domestic firm has an incentive to deviate from symmetric PCP to LCP if and only if:

1 2 V a r ( s t ) ( 2 α − 1 ) + ( 1 − α ) C o v ( s t , ν ~ t ) + α C o v ( s t , ν ~ t * ) − C o v ( s t , a t ) > 0

(65)

where, s t = log ( S t ) , ν ~ t = log ( ν t ) , ν ~ t * = log ( ν t * ) and a t = log ( A t ) .

Proof.

The condition follows from comparing π H t E d e v ( i ) > π H t E under lognormal distributions. Taking logs of the profit ratio yields the variance-covariance structure shown above.

Deviation from Symmetric LCP:

If all other firms use LCP, the deviating firm's profit is:

π H t I d e v ( i ) = E t − 1 [ β λ t λ t − 1 { [ p H t I d e v ( i ) − M C t ] n ( p H t I d e v ( i ) P H t I ) − θ P t * C t * P H t I } ]

(66)

The optimal price satisfies:

p H t I d e v ( i ) = θ θ − 1 E t − 1 [ P t * C t * λ t * S t θ − 1 M C t ] E t − 1 [ P t * C t * λ t * S t θ − 1 ]

(67)

Proposition 4

(Deviation from LCP): A domestic firm has an incentive to deviate from symmetric LCP to PCP if and only if:

1 2 V a r ( s t ) ( 2 α − 1 ) + [ ( 1 − α ) C o v ( s t , ν ~ t ) + α C o v ( s t , ν ~ t * ) − C o v ( s t , a t ) ] < 0

(68)

With all variables log-normal, logging the inequality reduces it to the trade-off underlying Proposition 4. What drives that trade-off is the variance of the nominal exchange rate and its covariance with factor prices ( ν ~ t , ν ~ t * ) and with productivity ( a t ) . How much the foreign input bill counts in the comparison is set by the import share, α . A positive covariance between exchange rate and factor prices works in LCP's favour: costs and revenues then move together, leaving the firm naturally hedged.

The foreign firms are handled by the same apparatus. With the two countries assumed perfectly symmetric, α serves as the import intensity for foreign and domestic firms equally, measuring how far each draws on the other's intermediates. Under log-normality this gives:

Proposition 5

(Foreign Firm Equilibria): The stable Nash equilibria for the foreign country are characterised as follows:

A symmetric LCP equilibrium (pricing in domestic currency) is stable if:

1 2 V a r ( s t ) ( 2 α − 1 ) − ( α C o v ( s t , ν ~ t ) + ( 1 − α ) C o v ( s t , ν ~ t * ) − C o v ( s t , a t * ) ) > 0

A symmetric PCP equilibrium (pricing in foreign currency) is stable if:

1 2 V a r ( s t ) ( 2 α − 1 ) − ( α C o v ( s t , ν ~ t ) + ( 1 − α ) C o v ( s t , ν ~ t * ) − C o v ( s t , a t * ) ) < 0

The conditions for foreign firms mirror those for domestic firms once the currency roles are flipped: pricing in domestic currency (LCP for a foreign firm) lines up with PCP seen from home, and conversely. The underlying logic is unchanged—the trade-off rests on the variance of the exchange rate and its covariances with factor prices, weighted by the import share α. What differs is the productivity term a t * , now foreign productivity, and a sign structure reversed to reflect the opposite exchange rate exposure foreign firms carry when they export.

4.3 The General Arbitrage Condition and Equilibrium Selection

The conditions derived in Propositions 3, 4, and 5 form the core of our general equilibrium analysis. They provide a unified arbitrage condition that determines the optimal invoicing strategy by balancing firm-level cost exposure against macroeconomic volatility. Comparing the deviation incentives in Propositions 3 and 4, we can identify the conditions for a stable Nash equilibrium in which all domestic firms coordinate on the same strategy. Proposition 6 states these conditions.

Proposition 6

(Stable Nash Equilibria): The stable Nash equilibrium for the domestic country is a symmetric outcome where all firms coordinate on the same invoicing strategy. A Producer Currency Pricing (PCP) equilibrium is stable if and only if:

1 2 V a r ( s t ) ( 2 α − 1 ) + ( 1 − α ) C o v ( s t , ν ~ t ) + α C o v ( s t , ν ~ t * ) − C o v ( s t , a t ) ≤ 0

(69)

A Local Currency Pricing (LCP) equilibrium is stable if and only if:

1 2 V a r ( s t ) ( 2 α − 1 ) + ( 1 − α ) C o v ( s t , ν ~ t ) + α C o v ( s t , ν ~ t * ) − C o v ( s t , a t ) ≥ 0

(70)

Note that these conditions are complementary: when one holds, the other does not, ensuring a unique symmetric equilibrium. The direction of the inequality depends on whether exchange rate variance or cost-revenue covariation dominates the firm's risk exposure.

This arbitrage condition is the central theoretical contribution of our general equilibrium model. It nests the partial equilibrium result from Proposition 2 as a special case. Specifically, if we assume that only the exchange rate varies (i.e., isolating the partial equilibrium effect by setting C o v ( s t , ν ~ t ) = C o v ( s t , ν ~ t * ) = C o v ( s t , a t ) = 0 , the condition simplifies to 1 2 V a r ( s t ) ( 2 α − 1 ) ≥ 0 for LCP stability. This recovers the threshold α > 0.5 from partial equilibrium: a firm with a production technology heavily reliant on imported goods ( α > 0.5 ) will opt for pricing in the buyer's currency (LCP).

General equilibrium opens a second channel, though, running through uncertainty in factor prices and productivity. Its covariances with the exchange rate reshape the plain threshold rule. A positive covariance between the exchange rate and factor prices ( C o v ( s t , ν ~ t ) > 0 , C o v ( s t , ν ~ t * ) > 0 ) , to take one case, nudges the firm towards LCP, because costs and revenues then climb and fall in tandem, each hedging the other. The covariance between the exchange rate and productivity runs the other way: ( C o v ( s t , a t ) > 0 ) positive, it favours PCP, since a depreciating domestic currency (higher s t ) coincides with higher productivity, lowering marginal costs, and boosting profits under PCP.

In that condition, the import share α serves as a weight, splitting the influence of foreign against domestic cost shocks. The invoicing choice is not down to any one factor. Three combine to produce it: the firm's cost structure (α), the variance of the exchange rate (Var(s_t)), and the way the exchange rate moves with economic fundamentals (Cov(s_t, ·)).

At the center of the general-equilibrium analysis is Proposition 6's arbitrage condition, valid for any joint distribution of exchange rate, factor prices, and productivity, and for any α. Propositions 7 and 8 are what is left after three additional restrictions: (A1) log utility in consumption, which links the stochastic discount factor to money supplies; (A2) random-walk money supplies, under which relative monetary variances alone drive the exchange rate; and (A3) monetary and productivity shocks that are independent, zeroing out every covariance term. Together, these three reduce the general condition to a contest between monetary variances—reproducing the monetary-stability finding of Devereux and Engel (2001).

4.4 The Role of Monetary Fundamentals Under Complete Markets

Getting closed-form solutions requires imposing some structure—particular functional forms plus a handful of independence assumptions. Remaining within the complete-markets setup, we have money supplies follow random walks:

m t = m t − 1 + ϵ m t ,

(71)

m t * = m t − 1 * + ϵ m * t

(72)

where m t = log ( M t ) , and productivity follows AR(1) processes:

a t = ρ a a t − 1 + ϵ a t ,

(73)

a t * = ρ a a t − 1 * + ϵ a * t

(74)

a I t = ρ a I a I , t − 1 + ϵ a I t ,

(75)

a I t * = ρ a I a I , t − 1 * + ϵ a I * t

(76)

For the baseline calibrations, we treat monetary and productivity shocks as uncorrelated. This is grounded in evidence: monetary policy and technology shocks seldom move together.

Under complete markets and log-normality, the international risk-sharing condition implies:

s t = m t − m t * + { terms\; involving\; productivity\; differences }

However, when monetary shocks dominate and are independent of productivity shocks, the covariance terms vanish because:

The exchange rate s t is primarily driven by relative money supplies ( m t − m t * )

Factor prices ( v t , v t * ) depend on productivity shocks through A I t , A I t *

By independence, C o v ( s t , v ~ t ) ≈ 0 , C o v ( s t , v ~ t * ) ≈ 0 , C o v ( s t , a t ) ≈ 0

Combining this with the factor price equations derived from optimisation:

v t = η β M t A I t ,

(77)

v t * = η β M t * A I t *

(78)

With independence imposed, the cumbersome covariance terms in Propositions 5–6 vanish, leaving only the monetary-uncertainty variance terms.

Proposition 7

(Domestic Choice Driven by Monetary Variance): Under the stated assumptions, all domestic firms will choose to:

Fix prices in the buyer's currency (LCP) if and only if V a r t − 1 ( m t ) > V a r t − 1 ( m t * )

Fix prices in their own currency (PCP) if and only if V a r t − 1 ( m t ) < V a r t − 1 ( m t * )

Proof.

When monetary shocks are independent of productivity shocks and dominate exchange rate movements, the covariance terms in Propositions 3–6 become negligible. The condition simplifies to depend only on relative monetary variances because C o v ( s t , ⋅ ) ≈ 0 for all factor price and productivity terms, and s t ≈ m t − m t * , so V a r ( s t ) ≈ V a r ( m t ) + V a r ( m t * ) .

Proposition 8

(Global Invoicing Equilibrium): There exists an equilibrium in which all firms in both countries fix their export prices in the currency of the country with the more stable monetary policy, i.e., the lower variance of money supply:

If V a r t − 1 ( m t ) > V a r t − 1 ( m t * ) ⇒ All firms use the foreign currency (LCP for domestic firms, PCP for foreign firms).

If V a r t − 1 ( m t ) < V a r t − 1 ( m t * ) ⇒ All firms use the domestic currency (PCP for domestic firms, LCP for foreign firms).

Proposition 8 is a limiting case of Proposition 6. Three conditions are jointly necessary: complete financial markets, monetary shocks independent of productivity shocks, and random-walk money supplies. Relaxing any one of the three breaks the simple monetary-variance ranking. What is new relative to Devereux and Engel (DE) (2001) is the general condition (Proposition 6) from which their result is derived, the identification of import intensity α as the structural weight governing the relative importance of the cost and monetary channels, and the formal characterisation of market completeness as the switching condition between the two regimes (Proposition 9, Section 4.5).

As established in Section 1, our supply-side cost-channel mechanism is complementary to the demand-side banking view of Gopinath and Stein (GS) (2021). When markets are complete, this exposure is fully hedgeable, and monetary stability dominates. When markets are incomplete, the exposure becomes a genuine unhedgeable risk, and the firm's production architecture determines the invoicing equilibrium. A unified framework incorporating both channels remains a productive avenue for future work. Table 1 contrasts the general arbitrage condition of Proposition 6 with the monetary-stability limiting case of Proposition 8.

OPEN IN VIEWER

Table 1.

General Condition vs. Limiting Case.

Dimension	Proposition 6 (General)	Proposition 8 (Special Case)
Financial markets	Complete (assumed for tractability)	Complete (required)
Monetary vs. productivity shocks	Correlated in general; C o v ( s , a ) unrestricted	Independent; C o v ( s , a ) = 0
Exchange rate process	General log-normal with full covariance structure	Random walk driven by M / M *
Import intensity α	Governs the weight of cost vs. the monetary channel	Subsumed — irrelevant in limiting case
Determines equilibrium	Full variance-covariance inequality (Proposition 6)	V a r ( m ) vs. V a r ( m * ) comparison only
Relationship to DE (2001)	Nests DE (2001) as a limiting case	Replicates DE (2001) exactly
Relationship to GS (2021)	Extends via cost-side channel and α	Does not capture the banking channel

This result, however, rests on the assumption of complete financial markets. Handy though that is, it presumes firms—or the households behind them—can hedge away the entirety of their exchange rate risk with state-contingent contracts. Markets do not actually work that way. They are incomplete, and the instruments a firm would need are frequently costly or unavailable, all the more so for the idiosyncratic risk it carries at its own level.

Incompleteness changes everything about the calculus, because a firm can no longer hedge at will through financial instruments. Now the covariance terms in the general arbitrage condition (Proposition 6) matter in earnest: they capture risks that resist hedging and have to be managed by operational means, the currency choice among them. The cost-shock exposure a firm picks up through imported inputs—summarised by α and the covariances—cannot be offset in the market any longer. With that, the microeconomic hedging motive returns to the foreground.

At this point, Propositions 7 and 8, built on complete markets, may no longer hold. Invoicing turns instead on the firm's own cost structure (α) and on the comovement of exchange rates with its input costs—exactly what the partial-equilibrium analysis (Proposition 2) and the general condition of Proposition 6 point to. Our model thus provides a clear theoretical bridge: the dominance of macroeconomic (monetary) factors over microeconomic (cost-structure) factors in invoicing decisions is not universal but conditional on the degree of financial market completeness. This conditional result motivates the formal analysis of incomplete markets that follows in the next Section.

The preceding results depend on the complete-markets assumption, under which households perfectly insure against all idiosyncratic risks through state-contingent financial contracts. This section relaxes that assumption by introducing a bond economy in which only a single non-contingent nominal bond B is traded internationally. This is the simplest form of market incompleteness and the most widely used tractable alternative in the open-economy literature (Betts & Devereux, 2000; Corsetti & Pesenti, 2001).

4.5.1 Structure of the Bond Economy

The household's budget constraint is modified to exclude state-contingent claims:

P t C t + M t + q t B t + 1 ≤ P t w t H t + M t − 1 + T t + B t + D t + ∫ π ( j ) d j

(79)

where q t is the bond price and B t + 1 is the quantity of the one-period nominal bond carried into period t + 1 . The critical departure from Section 4.1.2 is that the international risk-sharing condition λ t * = S t λ t , which underpins Propositions 7 and 8, no longer holds in general. Instead, the first-order condition for bond holdings yields:

q t = β E t [ λ t + 1 λ t ]

(80)

which equates the bond price to the expected discounted domestic marginal utility. Because foreign households face an analogous condition in foreign currency, the exchange rate and the marginal utilities are now jointly determined by both real (productivity) and monetary shocks, and the covariance terms in Proposition 6 are no longer driven exclusively by monetary variances.

4.5.2 The Invoicing Condition Under Incomplete Markets

Under the bond economy, the general arbitrage condition of Proposition 6 continues to hold: LCP is stable iff:

1 2 V a r ( s ) ( 2 α − 1 ) + ( 1 − α ) C o v ( s , ν ) + α C o v ( s , ν * ) − C o v ( s , a ) ≥ 0

(81)

but the covariance terms now contain real components that cannot be offset by financial markets. Specifically, let the exchange rate under incomplete markets be denoted s t B (superscript B for bond economy). Because risk-sharing is imperfect, s t B depends on both monetary shocks ( ε m , ε m * ) and productivity shocks ( ε a , ε a * ), and the orthogonality conditions C o v ( s , ν ) ≈ 0 and C o v ( s , a ) ≈ 0 that were imposed under complete markets in Section 4.4 no longer hold. The formal derivation is provided in Appendix C. This yields the following result:

Proposition 9

(Invoicing Choice under Incomplete Markets): In the bond economy, the stable Nash invoicing equilibrium is determined by the general condition of Proposition 6 with non-zero covariance terms. In particular:

When α > 1 2 and the covariance between the exchange rate and imported input prices, C o v ( s t B , ν t * ) , is positive — which occurs when both are driven by common real shocks — the LCP equilibrium is stable irrespective of the relative variances of monetary policy. The micro-hedging motive dominates.

When α < 1 2 and C o v ( s t B , ν t * ) is small or negative, the PCP equilibrium is stable, and the monetary stability ranking of Proposition 8 provides a useful approximation only to the extent that monetary shocks remain the dominant driver of exchange rate variance.

As financial markets approach completeness and risk-sharing improves, Proposition 8's macro-stability result re-emerges as a limiting case—which confirms what it is: a feature of the complete-markets special case, not a general law.

Proof.

See Appendix C.