# The Cashless Economy With Two Countries Essay

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II. Model
Consider a cashless economy with two countries, each with individual fiscal authority and the central bank. A representative household in one country seeks to maximize her lifetime utility by choosing the amount of consumption
max┬{c_it^a,c_it^b } ⁡〖E_0 ∑_(t=0)^∞▒〖β^t u(c_it^a,c_it^b)〗〗 (1)
for i=1,2, which is the index for households in each country, and 0<1. c_it^a represents the goods producing in country a and consumed by the household in country i. Assume that consumption goods in both countries are perfect substitutes.
The governments in both countries collect lump-sum taxes from the households, and they issue one-period bond, which can be redeemed at I_t^j (j=a,b) at time t+1 for one unit bond issued at time t. The households can choose to hold bonds from either government, so the household in country a faces a constraint at
p_t^a c_1t^a+e_t p_t^b c_1t^b+B_1t^a+e_t B_1t^b≤B_(1t-1)^a I_(t-1)^a+e_t B_(1t-1)^b I_(t-1)^b+p_t^a y_1t^a+e_t p_t^b y_1t^b-p_t^a T_t^a (2)
Here y_1t^a and y_1t^b denote the household 1’s real holding of goods from country a and country b, T_t^a is the real lump-sum tax paid in period t, and the exchange rate is given by e_t.
The budget constraint for the household in country b is symmetric, so solving for the equilibrium condition of one household will lead to a symmetric result for the other. We focus on country 1’s household. In order for that there is no arbitrage condition in the economy, interest rate parity condition must hold
e_t=(I_t^b)/(I_t^a ) E_t e_(t+1) (3)

Iterate (2) forward with (3), and the household faces the intertemporal budget const...

... middle of paper ...

...ere is one important assumption that I did not write down explicitly. For the above model to work, I assume that one country cannot run a Ponzi scheme on another country. Canzoneri, Cumby and Diba (2001) have already explained the importance of such an assumption. If I allow one country to borrow indefinitely from another, there will be no solution of either prices or exchange rate as Dupor (1999) proved. The two countries’ government and central bank present value budget constraints combine to one, and as long as the composite PVBC is satisfied, government surpluses can split into any ratio and borrowing does not matter. The joint PVBC is unable to pin down both countries’ prices. In my model, because of the no Ponzi scheme assumption, PVBCs determine price levels and the exchange rate in both countries. Consumptions are subsequently pinned down through the prices.

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