Microeconomic Foundation for Phillips Curve with a Three-Period Overlapping Generations Model and Negative Real Balance Effect

We use the following notations.

Consumers in period 0 consume the goods by borrowing money from consumers of the previous generation or the government (e.g. scholarship). They must repay the debts when they are young. However, if they are unemployed, they cannot repay the debts. Then, they receive the unemployment benefits, which are covered by taxes on employed younger generation consumers. Thus, employed younger generation consumers must pay the taxes for unemployment benefit as well as they must repay their own debts. R and Θ satisfy the following relation. (1) (Lf−L)R=LΘ. \left( {{L_f} - L} \right)R = L\Theta . In addition, we consider the existence of pay-as-you-go pension for consumers of the older generation. They are also covered by taxes on employed consumers of the younger generation. Θ and ψ satisfy (2) LfΦ=LΨ. {L_f}\Phi = L\Psi . Consumptions in period 0 of the consumers are constant, and in period 1, they determine their consumptions in periods 1 and 2 and their labour supply. δ is the definition function. If a consumer is employed, then δ = 1; if he is not employed, then δ = 0. The labour productivity is y(L). We consider increasing or constant returns to scale technology. Thus, y(L) is increasing or constant with respect to the employment of a firm L. We define the employment elasticity of the labour productivity as follows: ζ=y′y(L)L. \zeta = {{y'} \over {{{y\left( L \right)} \over L}}}. 0 ≤ ζ < 1, and it is constant. Increasing returns to scale means η > 0. η is (the inverse of) the degree of differentiation of the goods. In the limit when η→+∞, the goods are homogeneous. h and ζ satisfy (1−1η)(1+ζ)<1 \left( {1 - {1 \over \eta }} \right)\left( {1 + \zeta } \right) < 1 so that the profits of firms are positive.

The utility of consumers of one generation over three periods is U(X0,X1,X2,δ,β)=u(X0,X1,X2)−δβ. U\left( {{X^0},{X^1},{X^2},\delta ,\beta } \right) = u\left( {{X^0},{X^1},{X^2}} \right) - \delta \beta . u(X⁰, X¹, X²) is homogeneous of degree one (linearly homogeneous) with respect to X¹ and X². Note that X⁰ is constant. The budget constraint is ∫01p1(z)c1(z)dz+∫01p2(z)c2(z)dz=δW+Π+Φ′−δR−δ(Θ+Ψ). \matrix{ {\int_0^1 {{p^1}\left( z \right){c^1}\left( z \right)dz + } \int_0^1 {{p^2}\left( z \right){c^2}\left( z \right)dz = } } \hfill \cr {\delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right).} \hfill \cr } p²(z) is the expectation of the price of good z in period 2. Note that R = X⁰. The Lagrange function is ℒ=u(X0,X1,X2)−δβ−λ(∫01p1(z)c1(z)dz+∫01p2(z)c2(z)dz−δW−Π−Φ′+δR+δ(Θ+Ψ)). \matrix{ {{\cal L} = u\left( {{X^0},{X^1},{X^2}} \right) - \delta \beta - \lambda \left( {\int_0^1 {{p^1}\left( z \right){c^1}\left( z \right)dz + } } \right.} \hfill \cr {\left. {\int_0^1 {{p^2}\left( z \right){c^2}\left( z \right)dz - } \delta W - \Pi - \Phi ' + \delta R + \delta \left( {\Theta + \Psi } \right)} \right).} \hfill \cr } λ is the Lagrange multiplier. The first-order conditions are (3) ∂u∂X1(∫01(c1 z)1−1ηdz)1η1−1ηc1(z)−1η=λp1(z) {{\partial u} \over {\partial {X^1}}}{\left( {\int_0^1 {{{\left( {{c^1}\,z} \right)}^{1 - {1 \over \eta }}}dz} } \right)^{{{{1 \over \eta }} \over {1 - {1 \over \eta }}}}}{c^1}{\left( z \right)^{{1 \over \eta }}} = \lambda {p^1}\left( z \right) and (4) ∂u∂X2(∫01(c2 z)1−1ηdz)1η1−1ηc2(z)−1η=λp2(z) {{\partial u} \over {\partial {X^2}}}{\left( {\int_0^1 {{{\left( {{c^2}\,z} \right)}^{1 - {1 \over \eta }}}dz} } \right)^{{{{1 \over \eta }} \over {1 - {1 \over \eta }}}}}{c^2}{\left( z \right)^{- {1 \over \eta }}} = \lambda {p^2}\left( z \right) They are rewritten as (5) ∂u∂X1X1(∫01c1(z)1−1ηdz)−1c1(z)1−1η=λp1(z)c1(z), {{\partial u} \over {\partial {X^1}}}{X^1}{\left( {\int_0^1 {{c^1}{{\left( z \right)}^{1 - {1 \over \eta }}}dz} } \right)^{ - 1}}{c^1}{\left( z \right)^1}^{ - {1 \over \eta }} = \lambda {p^1}\left( z \right){c^1}\left( z \right), (6) ∂u∂X2X2(∫01(c2 z)1−1ηdz)−1c2(z)1−1η=λp2(z)c2(z) {{\partial u} \over {\partial {X^2}}}{X^2}{\left( {\int_0^1 {{{\left( {{c^2}\,\,z} \right)}^{1 - {1 \over \eta }}}dz} } \right)^{ - 1}}{c^2}{\left( z \right)^1}^{ - {1 \over \eta }} = \lambda {p^2}\left( z \right){c^2}\left( z \right) Let P1=(∫01p1(z)1−ηdz)11−η, P2=(∫01p2(z)1−ηdz)11−η. {P^1} = {\left( {\int_0^1 {{p^1}{{\left( z \right)}^{1 - \eta }}dz} } \right)^{{1 \over {1 - \eta }}}},\,{P^2} = {\left( {\int_0^1 {{p^2}{{\left( z \right)}^{1 - \eta }}dz} } \right)^{{1 \over {1 - \eta }}}}. They are price indices of the consumption baskets in periods 1 and 2. By some calculations, we obtain (see Appendix A) (7) u(X0,X1,X2)=λ[∫01p1(z)c1(z)dz+∫01p2(z)c2(z)dz]=λ[δW+Π+Φ′−δR−δ(Θ+Ψ)], \matrix{ {u\left( {{X^0},{X^1},{X^2}} \right) = \lambda \left[ {\int_0^1 {{p^1}\left( z \right){c^1}\left( z \right)dz + \int_0^1 {{p^2}\left( z \right){c^2}\left( z \right)dz} } } \right]} \hfill \cr { = \lambda \left[ {\delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right)} \right],} \hfill \cr } P2P1=∂u∂X2∂u∂X1 {{{P^2}} \over {{P^1}}} = {{{{\partial u} \over {\partial {X^2}}}} \over {{{\partial u} \over {\partial {X^1}}}}} and (8) P1X1+P2X2=δW+Π+Φ′−δR−δ(Θ+Ψ). {P^1}{X^1} + {P^2}{X^2} = \delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right). The indirect utility of consumers is written as follows: (9) V=1φ(P1,P2)[δW+Π+Φ′−δR−δ(Θ+Ψ)]−δβ. V = {1 \over {\varphi \left( {{P^1},{P^2}} \right)}}\left[ {\delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right)} \right] - \delta \beta . ϕ(P¹, P²) is a function which is homogeneous of degree one. The reservation nominal wage rate W^R is a solution of the following equation. 1φ(P1,P2)[WR+Π+Φ′−R−Θ−Ψ]−β=1φ(P1,P2)(Π+Φ′). \matrix{ {{1 \over {\varphi \left( {{P^1},{P^2}} \right)}}\left[ {{W^R} + \Pi + \Phi ' - R - \Theta - \Psi } \right] - \beta = } \hfill \cr {{1 \over {\varphi \left( {{P^1},{P^2}} \right)}}\left( {\Pi + \Phi '} \right).} \hfill \cr } From this, we have WR=φ(P1,P2)β+R+Θ+Ψ. {W^R} = \varphi \left( {{P^1},{P^2}} \right)\beta + R + \Theta + \Psi . The labour supply is indivisible. If W > W^R, the total labour supply is L_f. If W < W^R, it is zero. If W = W^R, then employment and unemployment are indifferent for consumers, and there exists no involuntary unemployment even if L < L_f.

Indivisibility of labour supply may be due to the fact that there exists minimum standard of living even in the advanced economy (Otaki, 2015).

Let ρ=P2P1 \rho = {{{P^2}} \over {{P^1}}} . This is the expected inflation rate (plus one). Since ϕ(P¹, P²) is homogeneous of degree one, the reservation real wage rate is ωR=WRP1=φ(1,ρ)β+R+Θ+ΨP1. {\omega ^R} = {{{W^R}} \over {{P^1}}} = \varphi \left( {1,\rho } \right)\beta + {{R + \Theta + \Psi } \over {{P^1}}}. If the value of ρ is given, ω^R is constant.

Otaki (2007) assumes that the wage rate equals the reservation wage rate in the equilibrium. However, there exists no mechanism to equalise them.

Let α=P1X1P1X1+P2X2=X1X1+ρX2,0<α<1. \alpha = {{{P^1}{X^1}} \over {{P^1}{X^1} + {P^2}{X^2}}} = {{{X^1}} \over {{X^1} + \rho {X^2}}},0 < \alpha < 1.

By some calculations, we obtain the demand for good z of a consumer of the younger generation as follows (see Appendix B): (10) c1(z)=α[δW+Π+Φ′−δR−δ(Θ+Ψ)]P1(p1(z)P1)−η. {c^1}\left( z \right) = {{\alpha \left[ {\delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right)} \right]} \over {{P^1}}}{\left( {{{{p^1}\left( z \right)} \over {{P^1}}}} \right)^{ - \eta }}. Similarly, his demand for good z in period 2 is c2(z)=(1−α)[δW+Π+Φ′−δR−δ(Θ+Ψ)]P2(p1(z)P2)−η. {c^2}\left( z \right) = {{\left( {1 - \alpha } \right)\left[ {\delta W + \Pi + \Phi ' - \delta R - \delta \left( {\Theta + \Psi } \right)} \right]} \over {{P^2}}}{\left( {{{{p^1}\left( z \right)} \over {{P^2}}}} \right)^{ - \eta }}. Let M be the total savings of consumers of the older generation carried over from their period 1. It is written as M=(1−α)[WL¯+LfΠ¯+LfΦ−L¯R¯−L¯(Θ¯+Ψ¯)]. M = \left( {1 - \alpha } \right)\left[ {\overline {WL} + {L_f}\bar \Pi + {L_f}\Phi - \bar L\bar R - \bar L\left( {\bar \Theta + \bar \Psi } \right)} \right]. W̅, L̅ and Π¯ \bar \Pi are nominal wage rate, employment and profit in the previous period, respectively. R̅, Θ¯ \bar \Theta and Ψ¯ \bar \Psi are the unemployment benefit, tax for the unemployment benefit and tax for the pay-as-you-go pension in the previous period, respectively. Note that Φ is the pay-as-you-go pension for a consumer of the older generation. M is the total savings or the total consumption of the older generation consumers including the pay-as-you-go pensions they receive in their period 2. It is the planned consumption that is determined in period 1 of the older generation consumers. Net savings is the difference between M and the pay-as-you-go pensions in their period 2, which is written as follows: M−LfΦ=(1−α)[WL¯+LfΠ¯−L¯R¯−L¯(Θ¯+Ψ¯)]−αLfΦ. \matrix{ {M - {L_f}\Phi = \left( {1 - \alpha } \right)\left[ {\overline {WL} + {L_f}\bar \Pi - \bar L\bar R - } \right.} \hfill \cr {\left. {\bar L\left( {\bar \Theta + \bar \Psi } \right)} \right] - \alpha {L_f}\Phi .} \hfill \cr } With this M, their demand for good z is MP1(p1(z)P1)−η. {M \over {{P^1}}}{\left( {{{{p^1}\left( z \right)} \over {{P^1}}}} \right)^{ - \eta }}. The government expenditure constitutes the national income as well as consumptions of younger and older generations. The total demand for good z is written as c(z)=YP1(p1(z)P1)−η. c\left( z \right) = {Y \over {{P^1}}}{\left( {{{{p^1}\left( z \right)} \over {{P^1}}}} \right)^{ - \eta }}. Y is the effective demand defined by Y=α[WL+LfΠ+LfΦ′−LR−L(Θ+Ψ)]+LfR′+G+M. \matrix{ {Y = \alpha \left[ {WL + {L_f}\Pi + {L_f}\Phi ' - LR - L\left( {\Theta + \Psi } \right)} \right] + } \hfill \cr {{L_f}R' + G + M.} \hfill \cr } G is the government expenditure other than the pay-as-you-go pensions and the unemployment benefits, and R′ is the consumption in the childhood period of consumers of the next generation (about this demand function, see Otaki (2007, 2009)). The total employment, the total profits and the total government expenditure are written as follows: ∫01Ldz=L,∫01Πdz=Π,∫01Gdz=G. \int_0^1 {Ldz = L,} \int_0^1 {\Pi dz = \Pi ,} \int_0^1 {Gdz = G.} We have ∂c(z)∂p1(z)=−ηYP1p1(z)−1−η(P1)−η=−ηc(z)p1(z). {{\partial c\left( z \right)} \over {\partial {p^1}\left( z \right)}} = - \eta {Y \over {{P^1}}}{{{p^1}{{\left( z \right)}^{ - 1 - \eta }}} \over {{{\left( {{P^1}} \right)}^{ - \eta }}}} = - \eta {{c\left( z \right)} \over {{p^1}\left( z \right)}}. From c (z) = Ly (L), ∂L∂p1(z)=1y(L)+Ly′∂c(z)∂p1(z). {{\partial L} \over {\partial {p^1}\left( z \right)}} = {1 \over {y\left( L \right) + Ly'}}{{\partial c\left( z \right)} \over {\partial {p^1}\left( z \right)}}. The profit of Firm z is π(z)=p1(z)c(z)−Wy(L)c(z). \pi \left( z \right) = {p^1}\left( z \right)c\left( z \right) - {W \over {y\left( L \right)}}c\left( z \right). P¹ is given for Firm z. Note that the employment elasticity of the labour productivity is ζ=y′y(L)L. \zeta = {{y'} \over {{{y\left( L \right)} \over L}}}. The condition for profit maximisation with respect to p¹ (z) is c(z)+[p1(z)−1−Ly′1y(L)+Ly′y(L)W]∂c(z)∂p1(z)==c(z)+[p1(z)−Wy(L)+Ly′]∂c(z)∂p1(z)=0. \matrix{ {c\left( z \right) + \left[ {{p^1}\left( z \right) - {{1 - Ly'{1 \over {y\left( L \right) + Ly'}}} \over {y\left( L \right)}}W} \right]{{\partial c\left( z \right)} \over {\partial {p^1}\left( z \right)}} = } \hfill \cr { = c\left( z \right) + \left[ {{p^1}\left( z \right) - {W \over {y\left( L \right) + Ly'}}} \right]{{\partial c\left( z \right)} \over {\partial {p^1}\left( z \right)}} = 0.} \hfill \cr } From this, we have p1(z)=Wy(L)+Ly′−c(z)∂c(z)∂p1(z)=W(1+ζ)y(L)+1ηp1(z). {p^1}\left( z \right) = {W \over {y\left( L \right) + Ly'}} - {{c\left( z \right)} \over {{{\partial c\left( z \right)} \over {\partial {p^1}\left( z \right)}}}} = {W \over {\left( {1 + \zeta } \right)y\left( L \right)}} + {1 \over \eta }{p^1}\left( z \right). Therefore, we obtain p1(z)=W(1−1η)(1+ζ)y(L). {p^1}\left( z \right) = {W \over {\left( {1 - {1 \over \eta }} \right)\left( {1 + \zeta } \right)y\left( L \right)}}. With increasing returns to scale, since ζ > 0, p¹ (z) is lower than that in a case without increasing returns to scale given the value of W.

Since the model is symmetric, the prices of all goods are equal. Then, P¹ = p¹ (z).

Hence (11) P1=W(1−1η)(1+ζ)y(L). {P^1} = {W \over {\left( {1 - {1 \over \eta }} \right)\left( {1 + \zeta } \right)y\left( L \right)}}. The real wage rate is ω=WP1=(1−1η)(1+ζ)y(L). \omega = {W \over {{P^1}}} = \left( {1 - {1 \over \eta }} \right)\left( {1 + \zeta } \right)y\left( L \right). Since ζ is constant, this is increasing or constant with respect to L.

The nominal aggregate supply of the goods equals WL+LfΠ=P1Ly(L). WL + {L_f}\Pi = {P^1}Ly\left( L \right). The nominal aggregate demand is α[WL+LfΠ+LfΦ′−LR−L(Θ+Ψ)]+LfR′+G+M=αP1Ly(L)+α[LfΦ′−LR−L(Θ+Ψ)]+LfR′+G+M \matrix{ {\alpha \left[ {WL + {L_f}\Pi + {L_f}\Phi ' - LR - L\left( {\Theta + \Psi } \right)} \right] + {L_f}R' + G + M} \hfill \cr { = \alpha {P^1}Ly\left( L \right) + \alpha \left[ {{L_f}\Phi ' - LR - L\left( {\Theta + \Psi } \right)} \right] + {L_f}R' + G + M} \hfill \cr } Since they are equal, P1Ly(L)=αP1Ly(L)+α[LfΦ′−LR−L(Θ+Ψ)]+LfR′+G+M. \matrix{ {{P^1}Ly\left( L \right) = \alpha {P^1}Ly\left( L \right) + \alpha \left[ {{L_f}\Phi ' - LR - } \right.} \hfill \cr {\left. {L\left( {\Theta + \Psi } \right)} \right] + {L_f}R' + G + M.} \hfill \cr } From Eqs (1) and (2), we have (Lf−L)R=LΘ, \left( {{L_f} - L} \right)R = L\Theta , and LfΦ=LΨ. {L_f}\Phi = L\Psi . Therefore, we get P1Ly(L)=αP1Ly(L)+α(LfΦ′−LfΦ−LfR)+LfR′+G+M. \matrix{ {{P^1}Ly\left( L \right) = \alpha {P^1}Ly\left( L \right) + \alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R} \right) + } \hfill \cr {{L_f}R' + G + M.} \hfill \cr } This means P1Ly(L)=α(LfΦ′−LfΦ−LfR)+LfR′+G+M1−α {P^1}Ly\left( L \right) = {{\alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R} \right) + {L_f}R' + G + M} \over {1 - \alpha }} In real terms (12) Ly(L)=α(LfΦ′−LfΦ−LfR)+LfR′+G+MP1(1−α), Ly\left( L \right) = {{\alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R} \right) + {L_f}R' + G + M} \over {{P^1}\left( {1 - \alpha } \right)}}, or (13) L=α(LfΦ′−LfΦ−LfR)+LfR′+G+MP1(1−α)y(L). L = {{\alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R} \right) + {L_f}R' + G + M} \over {{P^1}\left( {1 - \alpha } \right)y\left( L \right)}}. If we consider the tax T for the government expenditure, Eq. (13) is as follows: (14) L=α(LfΦ′−LfΦ−LfR−T)+LfR′+G+MP1(1−α)y(L). L = {{\alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R - T} \right) + {L_f}R' + G + M} \over {{P^1}\left( {1 - \alpha } \right)y\left( L \right)}}. 11−α {1 \over {1 - \alpha }} is a multiplier. Eqs (13) and (14) mean that the employment L is determined by g+m. It cannot be larger than L_f. However, it may be strictly smaller than L_f (L < L_f). Then, there exists involuntary umemployment. Since the real wage rate ω=(1−1η)(1+ζ)y(L) \omega = \left( {1 - {1 \over \eta }} \right)\left( {1 + \zeta } \right)y\left( L \right) is increasing or constant with respect to L, and the reservation real wage rate ω^R is constant, if ω > ω^R there exists no mechanism to reduce the difference between them.

When the nominal wage rate falls, the price of the goods (price of consumption basket) falls. If the employment changes, the rate of a fall in the nominal wage rate and that of the price may be different in the case of increasing returns to scale. The real values of G, T, Φ, Φ′, R′ will not change even when the price of the goods falls. On the other hand, the nominal values of R and M-L_fΦ will not change. Then, whether the aggregate demand increases or decreases when the nominal wage rate and the price of the goods fall depend on whether M−LfΦ−αLfR M - {L_f}\Phi - \alpha {L_f}R is positive or negative. If M−L_fΦ−αL_fR<0, there is a negative real balance effect (or Pigou effect)

About the traditional discussion of real balance effects, refer Pigou (1943) and Kalecki (1944).

Suppose that the nominal wage rate falls in a period, for example, period t. From Eq. (11), the price of the goods also falls. Let P_t and P_t−1 be the price of the goods (price of the consumption basket) in period t and that in period t−1. The inflation rate from period t−1 to t, PtPt−1−1 {{{P_t}} \over {{P_{t - 1}}}} - 1 , falls given P_t−1. If the negative real balance effect works, the real aggregate demand decreases. Then, the output and the employment decrease. Therefore, the lower inflation rate is accompanied by employment loss, that is, higher unemployment rate.

Alternatively, suppose that the nominal wage rate rises. The price of the goods also rises. If the negative real balance effect works, the real aggregate demand increases. Then, the output and the employment increase. Therefore, the higher price or higher inflation rate is accompanied by an increase in employment. Thus, we obtain a negative relationship between the inflation rate and the unemployment rate (positive relationship between the inflation rate and employment) as represented by the Phillips curve. We have shown a negative relationship between the inflation rate and the unemployment rate in the same period given the price in the previous period. Figure 1 depicts an example of the Phillips Curve. U_t denotes the unemployment rate in period t. Figure 1 implies that our model explains the inflation rate between periods t−1 and t by the unemployment rate in period t.

Fig. 1

Let T be the tax revenue other than the taxes for the pay-as-you-go pension system and the unemployment benefits. Then, the budget constraint of the government is G=T, and the aggregate demand is (15) αP1Ly(L)+α[LfΦ′−LR−L(Θ+Ψ)−T]+LfR′+G+M with G=T. \matrix{ {\alpha {P^1}Ly\left( L \right) + \alpha \left[ {{L_f}\Phi ' - LR - L\left( {\Theta + \Psi } \right) - T} \right] + } \hfill \cr {{L_f}R' + G + M\,{\rm{with}}\,G = T.} \hfill \cr } With this, the aggregate demand Eq. (14) is (16) L=α(LfΦ′−LfΦ−LfR−T)+LfR′+G+MP1(1−α)y(L) L = {{\alpha \left( {{L_f}\Phi ' - {L_f}\Phi - {L_f}R - T} \right) + {L_f}R' + G + M} \over {{P^1}\left( {1 - \alpha } \right)y\left( L \right)}} This implies that the balanced budget multiplier is one.

Given labour productivity y(L) and nominal wage rate W, the price of the goods P¹ is determined by Eq. (11). If the government expenditure G increases given T, that is, an increment of the government expenditure is financed by seigniorage, from Eq. (16) employment L increases given the price P¹. Then, the Phillips curve in Figure 1 shifts to the left as shown in Figure 2, the employment increases and the unemployment rate decreases given the inflation rate. With increasing returns to scale, y(L) is increasing with respect to L and the prices of the goods fall. However, employment will still increase.

Fig. 2

The money supply will increase by the difference between government expenditure and the tax. If the increase in the government expenditure is financed by seigniorage, it equals the increase in money supply. Therefore, the fiscal policy in this article is also a monetary policy. It should be called a fiscal-monetary policy. The increase in money supply does not raise the price. Thus, money is not neutral.

Otaki and Tamai (2012) suppose that the low (or high) unemployment rate in a period, for example, period t−1 raises (or lowers) the labour productivity in period t by learning effect. If the unemployment rate in period t−1 increases, the labour productivity in period t falls. Then, from Eq. (11), the price of the goods rises, and the inflation rate from period t to period t+1 falls given the (expected) price of the goods in period t+1. Alternatively, a decrease in the unemployment rate in period t−1 raises the labour productivity in period t. Then, the price of the goods falls, and the inflation rate from period t to period t+1 rises given the (expected) price of the goods in period t+1. Thus, they have shown the negative relation between the unemployment rate in period t−1 and the inflation rate from period t to period t+1, Pt+1Pt−1 {{{P_{t + 1}}} \over {{P_t}}} - 1 . On the other hand, a fall in the price in period t means that the inflation rate from period t−1 to period t falls, that is, the low unemployment rate in period t−1 explains the low (not high) inflation rate from period t−1 to period t, PtPt−1−1 {{{P_t}} \over {{P_{t - 1}}}} - 1 .

Their Phillips curve is depicted in Figure 3. U_t–1 denotes the unemployment rate in period t−1.

Fig. 3

The thicker curve in Figure 3 means that the low unemployment rate in period t−1 explains the high inflation rate between periods t and t+1 in the model of Otaki and Tamai (2012). The thinner curve shows that the increase in the unemployment rate in period t−1 leads to the higher inflation rate between periods t−1 and t in their model.