Monetary/Fiscal Interactions: Cash in Advance

Monetary/Fiscal Interactions: Cash in Advance Behzad Diba University of Bern April 2011 (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 1 / 11

Stochastic Exchange Economy We consider next a closed economy with a stochastic endowment of a perishable good that can be consumed by households or by the government (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 2 / 11

Stochastic Exchange Economy We consider next a closed economy with a stochastic endowment of a perishable good that can be consumed by households or by the government In contrast to our small-open-economy model, this model focuses on the determination of interest rates in a stochastic monetary economy Ricardian Equivalence here says: given the path of government purchases, the financing decision (debt versus taxes) has no effect on the real interest rate (the assertions about consumption and output hold in a trivial way) (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 2 / 11

Stochastic Exchange Economy We consider next a closed economy with a stochastic endowment of a perishable good that can be consumed by households or by the government In contrast to our small-open-economy model, this model focuses on the determination of interest rates in a stochastic monetary economy Ricardian Equivalence here says: given the path of government purchases, the financing decision (debt versus taxes) has no effect on the real interest rate (the assertions about consumption and output hold in a trivial way) A simple way to introduce money is through a Cash-in-Advance (CIA) constraint (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 2 / 11

Timing and Goods Exchange At the beginning of each period, households observe the state of the economy consisting of their endowment (y t ) of a single perishable good, government purchases (G t ), a lump-sum tax (τ t ), and the money stock (M t+1 ) Next, they trade money and bonds in a financial exchange Then, the financial exchange closes, and households buy goods from each other with cash There is a unit mass of identical households (who don t like the "color" of their endowment), and each household consists of a seller and a shopper (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 3 / 11

Timing and Goods Exchange At the beginning of each period, households observe the state of the economy consisting of their endowment (y t ) of a single perishable good, government purchases (G t ), a lump-sum tax (τ t ), and the money stock (M t+1 ) Next, they trade money and bonds in a financial exchange Then, the financial exchange closes, and households buy goods from each other with cash There is a unit mass of identical households (who don t like the "color" of their endowment), and each household consists of a seller and a shopper In the goods exchange, shopper h faces the CIA constraint P t C h t M h t (1) (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 3 / 11

Financial Exchange Household h enters the financial exchange of time t with bonds and money acquired at t 1 and faces the budget constraint M h t + Bh t+1 (1 + i t ) + P t τ t M t + (M h t 1 P t 1 C h t 1) + B h t, (2) where τ t is a lump-sum tax and, as we will see, M t = P t 1 y t 1 represents the cash proceeds from sales in the goods exchange of t 1 We can formally show that the CIA constraint (1) is binding if i t > 0 (but it is good enough if you just understand the intuition for this) and write (2) as P t C h t + Bh t+1 (1 + i t ) + P t τ t M t + B h t (3) (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 4 / 11

Household FOCs The representative household maximizes [ ] E t β j u(ct+j h ) j=0 where 0 < β < 1, u (.) > 0, and u (.) < 0, subject to (3) (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 5 / 11

Household FOCs The representative household maximizes [ ] E t β j u(ct+j h ) j=0 where 0 < β < 1, u (.) > 0, and u (.) < 0, subject to (3) Letting Λ t denote the Lagrange multiplier on (3), the FOCs for Ct h and Bt h are u (Ct h ) = P t Λ t and the Euler equation Λ t = β(1 + i t )E t Λ t+1 (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 5 / 11

Household FOCs The representative household maximizes [ ] E t β j u(ct+j h ) j=0 where 0 < β < 1, u (.) > 0, and u (.) < 0, subject to (3) Letting Λ t denote the Lagrange multiplier on (3), the FOCs for Ct h and Bt h are u (Ct h ) = P t Λ t and the Euler equation Λ t = β(1 + i t )E t Λ t+1 These FOCs imply the Keynes-Ramsey rule { u (Ct h ) u (Ct+1 h = β(1 + i t )E ) } t P t P t+1 (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 5 / 11

Government and Equilibrium CIA models, in general, may or may not impose the CIA constraint on government purchases; we will impose it as P t G t M g t (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 6 / 11

Government and Equilibrium CIA models, in general, may or may not impose the CIA constraint on government purchases; we will impose it as In equilibrium, we have P t G t M g t C h t = C t h and C t + G t = y t and M h t + M g t = M t+1 (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 6 / 11

Government and Equilibrium CIA models, in general, may or may not impose the CIA constraint on government purchases; we will impose it as In equilibrium, we have and P t G t M g t C h t = C t h and C t + G t = y t M h t + M g t = M t+1 The model may treat government bonds as nominal or real (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 6 / 11

Government and Equilibrium CIA models, in general, may or may not impose the CIA constraint on government purchases; we will impose it as In equilibrium, we have and P t G t M g t C h t = C t h and C t + G t = y t M h t + M g t = M t+1 The model may treat government bonds as nominal or real With nominal bonds, the government budget constraint is M t+1 + B t+1 (1 + i t ) = M t + B t + P t (G t τ t ), with B t = B h t, in equilibrium (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 6 / 11

Ricardian Equivalence We have (trivially) the assertion about consumption: C t = y t G t (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 7 / 11

Ricardian Equivalence We have (trivially) the assertion about consumption: C t = y t G t If we add a real bond, we get the Euler equation u (C t ) = β(1 + r t )E t { u (C t+1 ) } and 1 1 + r t = βe t { u } (y t+1 G t+1 ) u (y t G t ) (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 7 / 11

Ricardian Equivalence We have (trivially) the assertion about consumption: C t = y t G t If we add a real bond, we get the Euler equation and u (C t ) = β(1 + r t )E t { u (C t+1 ) } 1 1 + r t = βe t { u } (y t+1 G t+1 ) u (y t G t ) We also get an analogous result for the real return on a nominal bond, up to a log-linear approximation (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 7 / 11

Government Budget in Equilibrium If we treat government bonds as nominal (just for now), and M t+1 + B t+1 (1 + i t ) = M t + B t + P t (G t τ t ), u (C t ) P t = β(1 + i t )E t { u } (C t+1 ) P t+1 imply a linear equation governing debt dynamics u { (C t )B t u = } [ (C t+1 )B t+1 βe t + u (C t ) τ t + M t+1 M t P t P t+1 P t with C t = y t G t, in equilibrium G t ] (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 8 / 11

Government PVBC Iterating the budget constraint forward and imposing the household s transversality condition, { u } (C t+j )B t+j lim j + βj E t = 0 we get the PVBC u (C t )B t P t = E t P t+j [ β {u j (C t+j ) τ t+j + M t+j+1 M t+j j=0 P t+j G t+j This equates real debt to the expected present value of primary surpluses (inclusive of seigniorage), calculated using the household s "stochastic discount factor" β j u (C t+j )/u (C t ) ]} (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 9 / 11

Another Rendition of Government PVBC Alternatively, we may write the budget constraint as M t+1 + B t+1 (1 + i t ) = M t + B t + P t (G t τ t ) i tm t+1 1 + i t, and define L t = M t + B t, to get the PVBC (as you can show for HOMEWORK) u (C t )L t P t = E t [ β {u j (C t+j ) τ t+j + i ]} t+jm t+j+1 G t+j j=0 (1 + i t+j ) P t+j (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 10 / 11

Another Rendition of Government PVBC Alternatively, we may write the budget constraint as M t+1 + B t+1 (1 + i t ) = M t + B t + P t (G t τ t ) i tm t+1 1 + i t, and define L t = M t + B t, to get the PVBC (as you can show for HOMEWORK) u (C t )L t P t = E t [ β {u j (C t+j ) τ t+j + i ]} t+jm t+j+1 G t+j j=0 (1 + i t+j ) P t+j This equates real public sector liabilities to the expected present value of primary surpluses inclusive of central bank transfers (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 10 / 11

CIA and a Simplification of Government PVBC The two renditions of the PVBC above apply with or without the CIA structure (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 11 / 11

CIA and a Simplification of Government PVBC The two renditions of the PVBC above apply with or without the CIA structure The CIA structure allows us to set M t+j+1 P t+j = y t+j and in the two versions of the PVBC M t+j+1 M t+j P t+j = y t+j ( Pt+j 1 P t+j ) y t+j 1 (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 11 / 11

CIA and a Simplification of Government PVBC The two renditions of the PVBC above apply with or without the CIA structure The CIA structure allows us to set M t+j+1 P t+j = y t+j and in the two versions of the PVBC M t+j+1 M t+j P t+j = y t+j ( Pt+j 1 P t+j For some applications, we may assume output and government purchases are constant; for example, the simple PVBC b t = B t = E t P t β {τ j t+j + j=0 ( Pt+j P t+j 1 P t+j ) } y G ) y t+j 1 will be suffi cient for our discussion of the "Monetarist Arithmetic" (Institute) Monetary/Fiscal Interactions: Cash in Advance April 2011 11 / 11