- An independent monetary policy.
It is both a hypothesis based on the Uncovered interest rate parity condition, and a finding from empirical studies where governments which have tried to simultaneously pursue all three goals have failed.
Theoretical derivation 
The formal model underlying the hypothesis is the uncovered Interest Rate Parity condition which states that in absence of a risk premium, arbitrage will ensure that the depreciation or appreciation of a country's currency vis a vis another will be equal to the nominal interest rate differential between them. Since under a peg, the exchange rate cannot change, short of devaluation or abandonment of the peg, this means that the two countries' nominal interest rates have to be equalized.
This in turn implies that the pegging country has no ability to set its nominal interest rate, and hence no independent monetary policy. The only way then that the country could have both a fixed exchange rate and an independent monetary policy is if it can prevent arbitrage in the foreign exchange rate market from taking place - institutes capital controls on international transactions.
Trilemma in practice 
The idea of the impossible trinity went from theoretical curiosity to becoming the foundation of open economy macroeconomics in the 1980s, by which time capital controls had broken down in many countries, and conflicts were visible between pegged exchange rates and monetary policy autonomy. While one version of the impossible trinity is focused on the extreme case – with a perfectly fixed exchange rate and a perfectly open capital account, a country has absolutely no autonomous monetary policy – the real world has thrown up repeated examples where the capital controls are loosened, resulting in greater exchange rate rigidity and less monetary-policy autonomy.
In 1997, Maurice Obstfeld and Alan M. Taylor brought the term "trilemma" into widespread use economics. In work with Jay Shambaugh, they developed the first methods to empirically validate this central, yet hitherto untested, hypothesis in international macroeconomics.
Economists Michael C. Burda and Charles Wyplosz provide an illustration of what can happen if a nation tries to pursue all three goals at once. To start with they posit a nation with a fixed exchange rate at equilibrium with respect to capital flows as its monetary policy is aligned with the international market. However, the nation then adopts an expansionary monetary policy to try to stimulate its domestic economy.
This involves an increase of the monetary supply, and a fall of the domestically available interest rate. Because the internationally available interest rate adjusted for foreign exchange differences has not changed, market participants are able to make a profit by borrowing in the country's currency and then lending abroad – a form of Carry trade.
With no capital control market players will do this en masse. The trade will involve selling the borrowed currency on the foreign exchange market in order to acquire foreign currency to invest abroad – this tends to cause the price of the nation's currency to drop due to the sudden extra supply. Because the nation has a fixed exchange rate, it must defend its currency and will sell its reserves to buy its currency back. But unless the monetary policy is changed back, the international markets will invariably continue until the government's foreign exchange reserves are exhausted, causing the currency to devalue, thus breaking one of the three goals and also enriching market players at the expense of the government that tried to break the impossible trinity.
Possibility of a dilemma 
In the modern world, given the growth of trade in goods and services and the fast pace of financial innovation, it is possible that capital controls can often be evaded. In addition, capital controls introduce numerous distortions. Hence, there are few important countries with an effective system of capital controls, though by early 2010 there has been a movement among economists, policy makers and the International Monetary Fund back in favour of limited use. Lacking effective control on the free movement of capital, the Impossible Trinity asserts that a country has to choose between reducing currency volatility and running a stabilising monetary policy: it cannot do both. As stated by Paul Krugman in 1999:
|“||The point is that you can't have it all: A country must pick two out of three. It can fix its exchange rate without emasculating its central bank, but only by maintaining controls on capital flows (like China today); it can leave capital movement free but retain monetary autonomy, but only by letting the exchange rate fluctuate (like Britain – or Canada); or it can choose to leave capital free and stabilize the currency, but only by abandoning any ability to adjust interest rates to fight inflation or recession (like Argentina today,).||”|
See also 
Notes and references 
- The Great Depression as a Watershed: International Capital Mobility in the Long Run, NBER Working Paper 5960, March 1997.. Retrieved 2012-12-20.
- The Trilemma in History: Tradeoffs among Exchange Rates, Monetary Policies, and Capital Mobility, NBER Working Paper 10396, March 2004.. Retrieved 2012-12-20.
- In real life examples, as the market players reach the point where they suspect the government is running out of the reserves to defend its currency, they will pile in with direct speculative attacks where they borrow and sell the nation's currency without bothering with carry trades, leading to a quick profit once the inevitable devaluation occurs.
- Michael C. Burda and Charles Wyplosz (2005). Macroeconomics: A European Text , 4th edition. Oxford University Press. pp. 246–248 , 515 , 516. ISBN 0-19-926496-1.
- Dani Rodrik (2010-03-11). "The End of an Era in Finance". Project Syndicate. Retrieved 2010-05-24.
- Kevin Gallagher (2010-03-01). "Capital controls back in IMF toolkit". The Guardian. Retrieved 2010-05-24.
- Arvind Subramanian (2009-11-18). "Time For Coordinated Capital Account Controls?". The Baseline Scenario. Retrieved 2009-12-15.
- Paul Krugman (1999-10-10). "O Canada – a neglected nation gets its Nobel.". Slate (magazine). Retrieved 2010-06-01.
- Please note that this was written in 1999, when Argentina had a Fixed exchange rate. Argentina abandoned this in January 2002. See Argentine economic crisis (1999–2002)