Household

Households’ preference depends on their consumption and labor supply. As is standard in New Keynesian models, we assume that consumption and labor supply are separable in the utility function. We assume further that the intertemporal elasticity of substitution of consumption is one. This value is within a standard range and commonly used in the real-business cycle and New Keynesian literature. When the utility function is separable, a unit intertemporal elasticity of substitution is a condition consistent with a balanced growth path. Therefore, the preference is represented by

where E_t denotes the expectation operator given information up to period t and β the discount factor. C_t is consumption and N_t labor supply in period t. C_t is an aggregate of Home product C_ht and Foreign product C_ft . The elasticity of substitution between Home and Foreign products is also assumed to be one. Due to this assumption, the monetary policy problem becomes more tractable and can be solved analytically. Indeed, Galí and Monacelli (2005) and Faia and Monacelli (2008) reported that the monetary policy that stabilizes domestic price is optimal in a small open economy under the elasticity assumption. Home and Foreign consumption aggregates are, respectively,

where ξ = (1−γ)η and ξ^* = γη for η ∈ [0,1] . η measures openness of the economies. When η is equal to zero, the two countries are closed economies and households consume domestic products only. In contrast, when η is equal to one, they are fully open and the consumption weights are the same with the countries’ sizes. When η is less than one, households are biased toward domestic products. In the limit case that γ goes to zero with positive η , Home is a small open economy and Foreign is a closed economy. In the literature, two country models sometimes abstract from home bias in consumption. Home bias is, however, essential in a small open economy model because, without the assumption, Home product is negligible in consumption baskets and thus both Home and Foreign agents do not care about the amount of Home production.

The elasticity of substitution among Home products and that among Foreign products are both θ > 1. That is, the aggregators of Home and Foreign products are, respectively,

Households’ budget constraint in state h^t is

where B_t+1(h^t+1) is the purchase of the bond that pays one unit of domestic currency in state h^t+1 and M_t(h^t+1) is the bond price. Households receive firms’ profits and government transfers.

Household h is a monopolistic competitive supplier of type-h labor and sets a wage rate W_t(h) in each period t. Labor input is aggregated as for wage markup ≥ 1 , which is assumed to be exogenous. We assume flexible wage so that it can be reset in each period. Every household faces the same problem of wage-setting, and the household optimization implies that the real wage in equilibrium is . That is, the wage is equal to the marginal value of labor supply, which is equal to the marginal rate of substitution between consumption and labor supply, multiplied by the wage markup. Fluctuation of plays a role in New Keynesian models as a cost-push shock, which implies a short-run trade-off between output and inflation stabilization. Similarly, is Foreign wage markup.

Firm

Home firm f ∈ [0,γ] produces with a constant-returns-to-scale technology Y_t(f) = A_tN_t(f) , where A_t is a country-specific productivity. Similarly, denotes Foreign firms’ productivity. Firms receive a subsidy τ for each employment (they pay a tax if τ < 0 .). Thus, firms’ nominal cost for unit employment is (1−τ)W_t . In the terms of modeling, it doesn’t matter whether firms or workers receive subsidy. Moreover, sales subsidy to firms or consumption subsidy to households for domestic products also yields the same conclusion.

Nominal rigidity is modeled by the assumption of a standard Calvo pricing technology that each firm cannot reset its price with probability α as introduced by Calvo (1983). The event is independent across firms and over time. We assume that a firm sets a single price in its own currency for both Home and Foreign markets. This implies that the law of one price holds for every individual product at all time, and , where the nominal exchange rate S_t is the Home currency price of Foreign currency.

Financial Markets

We assume that financial market is complete. We will, however, show that the financial market completeness is irrelevant with a certain initial condition.

We assume that disturbances (A_t, ) t t A m and (, ) t t A m follow Markov processes. This assumption is not critical but necessary to write the monetary policy problem in a recursive form.

Monetary Policy Problem in Closed Economies

If ξ = 0 , then Home is a closed economy. The consumer price and domestic price are identical and so are inflations, Π_ht = Π_t . Therefore, the monetary policy problem is

Cooperative Monetary-policy Problem

The monetary policy makers in the two countries maximize the weighted sum of the expected utilities of the two countries. We assume that every individual is equally weighted. That is, the weights of the two countries are proportional to the countries’ sizes, γ and 1 − γ . Then the objective function is

where we used γ(1 − ξ) + (1 − γ)ξ* = γ and γξ + (1 − γ)(1 − ξ*) = 1 − γ . Note that Home and Foreign variables in the objective function are additively separable. Also, the set of equilibrium conditions are divided into equations with Home variables and with Foreign variables. Thus, the optimal cooperative monetary policy should solve

subject to (1), (2), (3), and (4). The only difference from the monetary policy problem in closed economies is domestic inflation rather than CPI inflation.

Proposition 1 (Cooperative monetary policy) The monetary policy problem in open economies is isomorphic to that in closed economies with a transformation Π_t → Π_ht .

Remark The key reason for the isomorphism is that Home marginal cost does not depend on Foreign variables explicitly in open economies. Firms’ price setting is summarized by K_t and F_t, from which we can see the optimal reset price is the weighted average of current and future marginal costs. In general, Foreign production may affects Home marginal cost through two channels. First, if Foreign output increases, Home output becomes relatively scarce and hence the terms-of-trade improves. Given Home consumption level, it reduces Home marginal cost. Second, if Foreign output increases, so does Home consumption. Then the marginal value of leisure increases and thus labor input becomes more expensive. Therefore, the wealth effect by a rise in Foreign output implies higher Home marginal cost. As reported in Clarida, Galí, and Gertler (2002), the two effects are exactly offset when the elasticity of substitution is one. Thus, Home marginal cost does not depend on Foreign output directly. The property holds regardless of distortions, which is the main factor for Proposition 1.

The result in Clarida, Galí, and Gertler (2002) can be directly implied by Proposition 1. Compared with Proposition 1, the paper assumed no distortions in a steady state and showed their results in an approximated problem. We could extend their results because we compared not the solutions to the policy problems but the policy problems themselves.

Non-cooperative Monetary-policy Problem

The monetary policy maker maximizes Home households’ expected utility given Foreign agents’ decision. Now we will show a transformation that links the monetary policy problem in open economies to that in closed economies. Let Ỹ_t ≡ (1 − ξ)^−1/(1+ϕ)Y_t . Then we may rewrite the objective function as

Note that the second term is independent of Home monetary policy. Thus, the problem of the non-cooperative policy maker can be rewritten as

where and . The transformed problem has the same structure with that of the monetary policy problem in closed economies.

Proposition 2 (Non-cooperative monetary policy) The monetary policy problem in open economies is isomorphic to that in closed economies with a transformation (Y_t, Φ_t, Π_t) → ((1 − ξ)^-1/(1+ϕ)Y_t, (1 − ξ)Φ_t, Π_ht) .

Compared to the problem in closed economies, there is an additional term in distortions (1 − ξ) . That is, the monetary policy maker in open economies behaves as if there is a distortionary employment subsidy ξ . In other words, Home output is produced too much due to the subsidy in the view point of the policy maker and thus should be reduced at optimum.

The welfare function, equation (8), shows that only (1 − ξ) portion of domestic output is consumed by Home households in equilibrium. Thus, the tradeoff between consumption and labor supply is different in open economies. In particular, labor supply in open economies should be less than that in closed economies. That is, the open-economy monetary policy should be more contractionary.

1. Stabilization in the Case of a Non-distorted Steady State.

First, in the cases of the monetary policy cooperation, Proposition 1 says that the only difference in monetary policies is inflations that the policy maker targets. Thus, without distortions in a steady state, the result that the monetary policy maker in closed economies balances inflation and output gap stabilization continues to hold in open economies in that the policy maker balances domestic inflation and output gap stabilization.

Now we move on to the cases of the non-cooperative monetary policy. Suppose that fiscal policy is set such that there is no distortion in a steady state, = 1 or (1 − ξ)(1 − τ)μ^pμ^w = 1. Note that the log deviations from the efficient output do not change by the transformation, ln Ỹ_t − ln Ỹ^e = ln Y_t − ln Y^e , where Y^e is output in the flexible price equilibrium. Therefore, if one expresses the monetary policy in terms of output gap and inflation, the solutions to the monetary policy problems in open and closed economies would be the same except that in open economies domestic inflation is used instead of consumer price inflation. Again, in equilibrium the monetary policy would balance domestic inflation and output gap stabilization. Galí and Monacelli (2005) assumed that the wage markup is always one in a small open economy. Remind that as γ goes to zero, Home is a small open economy and ξ = (1 − γ)η → η . The paper further assumed that (1 − η)(1 − τ)μ^p = 1. Then for all t. In closed economies without a cost-push shock, price stabilization is an optimal monetary policy. According to the transformation in this paper, the open-economy counterpart is that domestic price stabilization is an optimal monetary policy, which is the same with the result in Galí and Monacelli (2005).

We have shown that in both cases, one can obtain the optimal monetary policy in open economies directly by applying the transformation in this paper. When there is no distortion in a steady state, the optimal monetary policy in open and closed economies are very similar. The subsidy rates for a non-distorted steady state are, however, different in open and closed economies. In the next subsection, we will show that the difference has an important implication.

2. Gains from Monetary Policy Cooperation

This paper claims that there exist gains from monetary policy cooperation since the non-cooperative monetary policy maker has an incentive to reduce domestic output. Clarida, Galí, and Gertler (2002) reported that there are no gains from monetary policy cooperation when the intertemporal elasticity of substitution is one. The main point of the paper is that with a unit elasticity of substitution Foreign output level does not affect Home marginal cost. An important assumption of the paper is that fiscal policy eliminates distortions in a steady state in both cooperative and non-cooperative cases. The required subsidy rates are, however, different in the two cases. We, hence, examine whether there exist gains from monetary policy cooperation given fiscal policy.

Proposition 3 Given fiscal policy, there are gains from monetary policy cooperation.

Proof. At optimum the welfare in the cooperative monetary policy cannot be less than that in the non-cooperative monetary policy. Then it is enough to show that the optimal non-cooperative monetary policy does not solve the problem of the cooperative monetary policy. The optimal non-cooperative monetary policy maximizes E₀ , while the optimal cooperative monetary policy does E₀ , where the constraints (1), (2), and (3) are common in the two cases. It is obvious that the solutions to the two problems are different, which completes the proof.

The non-cooperative policy maker induces less output than the cooperative policy maker does. As we have already explained, the reason is that the non-cooperative policy maker ignores Foreign consumption of domestic output. Corsetti and Pesenti (2001, 2005) also reported that the monetary policy maker in open economies has an incentive to manipulate the terms-of-trade. By more contractionary monetary policy, domestic outputs become more expensive and domestic households supply less labor. Although domestic consumption is also reduced, the reduction of labor supply has larger effect on the welfare. Therefore, the monetary policy maker tries to revalue the domestic currency.

The difference from the early literature is that we compare the welfares for given fiscal policy. The main lesson is that in open economies the monetary policy makers have a distorted incentive, which can be eliminated by the international monetary cooperation.

3. Inflation Bias

Following from the seminal papers of Kydland and Prescott (1977) and Barro and Gordon (1983), a large literature has studied the problem of inflation bias under discretionary or time-consistent monetary policy. This paper compares inflation bias in open and closed economies. This exercise will illustrate the differences of monetary policies in open and closed economy more clearly. Also, through the exercise one can see how to apply the transformation in this paper.

First, remind that the values of consumption and output are always the same, P_tC_t = P_htY_t . In terms of inflations, we have Π_t = Π_htC_t-1Y_t / (C_tY_t-1) . Therefore, the consumer price inflation and domestic inflation are the same in stationary equilibrium, Π = Π_h . That is, it is not necessary to distinguish the two inflations in stationary equilibrium.

The optimal inflation rate in stationary equilibrium of the Ramsey problem is zero (Π = 1) as shown in Benigno and Woodford (2005). Now consider the optimal inflation rate in stationary equilibrium of the time-consistent policy problem. Since we are focusing on stationary equilibrium, exogenous aggregate variables are assumed to have their steady state values. The equilibrium concept here is a Markov perfect equilibrium. As in Klein, Krusell, and Ríos-Rull (2008), a time-consistent equilibrium consists of a value function V (·) and policy functions {Ỹ(Δ), K(Δ), F(Δ), Δ'(Δ)} such that for all Δ ≥ 1, the policy functions solve

and the value function satisfies

One may think this problem as the monetary policy maker in each period chooses current monetary policy given the future policy function. That is, the monetary policy maker cannot commit future policy. Nonetheless, monetary policy in each period affects future policy through the state variable Δ . The monetary policy maker rationally expects that the future monetary policy is also optimal given price dispersion in the future period. The allocations in a stationary equilibrium satisfy Δ = Δ'(Δ) , K = K(Δ) , F = F(Δ) . Then the inflation in the stationary equilibrium is Inflation bias is, therefore, .

In a closed economy model, Woodford (2003) showed that when distortion is small (that is, ), inflation bias is

According to the transformation, inflation bias in open economies is

Proposition 4 Inflation bias is decreasing in openness in the case of small distortions.

Although we conjecture that the proposition holds even in the case of large distortions, we have not proved it yet. Proposition 4 is in line with the empirical result in Romer (1993), which reported that average inflation was lower in more open economies.

Corsetti and Pesenti (2001, 2005) also reported the difference in open and closed economies and less inflationary bias in open economies. The papers, however, examined how unanticipated monetary shocks affect the welfare. The exercise in the papers is useful to understand intuition but inconsistent with rational expectation equilibrium. In their model, agents set price one period in advance. Anticipated inflations do not have any effect on the allocations and thus welfare. That is, if agents understand the policy maker’s incentive and expect contractionary monetary policy, they will change prices accordingly. Indeed, there is no time-consistent rational expectation equilibrium when the monetary policy maker has such an incentive. In contrast, we model nominal rigidity by a Calvo-pricing technology and obtain inflation bias in rational expectation equilibrium.

For calibration we mostly follow Christiano, Eichenbaum, and Evans (2005), which is a key paper in the literature on monetary policy estimation. We let the period length be a quarter and the discount factor β =1.03^-0.25 , which implies an annualized real interest rate of 3%. The inverse of the Frisch elasticity of labor supply is set to be one, ϕ = 1. The Calvo-pricing parameter is α = 0.6 , which means that firms change their prices every 2.5 quarters on average. Price and wage markups are μ^p =1.2 and μ^w =1.05 .

Christiano, Eichenbaum, and Evans (2005) studied a closed economy and abstracted from fiscal policy. In open economies, we need to calibrate the weight of imports in the consumption basket ξ . The volume of international trade has been growing faster than world production. A reason is the globalization of supply chains as reported in Hummels, Ishii, and Yi (2001). As each country has specialized in particular production stages, trades of intermediate goods have increased rapidly. In such supply chains, a fraction of imports are used not for domestic consumption but for export. For our purpose, we may have to exclude imports for export. In the model in this paper, the values of consumption and output are the same. Hence, ξ is the ratio of imports to GDP, where imports should include only goods and services for domestic use. For calibration we have chosen the Korean economy. The value of imports to Korea is about a half of the value of GDP of Korea. Around 60 percent of imports are for domestic use. Thus, we set ξ = 0.3 .

The remaining parameter is the employment subsidy τ . Note that it does not matter whether firms or workers receive subsidy and that sales subsidy to firms or consumption subsidy to households for domestic products also yields the same conclusion. The thing is the overall distortion by fiscal policy. To calibrate the subsidy, we use the total tax revenue (excluding social security) in the OECD database. The tax revenue of the Korean economy in 2012 is about 20.2% of GDP, which is a little higher than the OECD average. Thus, we set τ = −0.202 .

Then the overall distortion is = 1.06 or 6%, which is far smaller than the corresponding distortion in closed economies, = 1.51 or 51%. With these parameters the annualized inflation bias in the open economy is estimated to be about 2.0%, which is far less than that in closed economies, 18.2%. That is, the monetary policy maker in the open economy has far smaller inflationary bias. Although the model in this paper is highly stylized and thus the estimates of inflation bias should be interpreted with caution, this example shows that the distortion from terms-of-trade manipulation may be as important as the distortions from price and wage markups.

When distortions are not small-enough, one may not use a perturbation method around an efficient equilibrium. In closed economy models Anderson, Kim, and Yun (2010) obtained inflation bias in a Markov perfect equilibrium using a projection method. The numerical method is also applicable to the problem in open economies by adjusting distortions.