Greece’s Financial Tragedy: Comparative Statics Using The Implicit Function Theorem and the Cramer Rule

At the time of these writings Greece is going through a severe sovereign debt crisis which is severely damaging the value of the Euro in relation to the dollar.  A mathematical model can be created to determine how a reduction in Greek government spending would impact its Gross Domestic Product, domestic interest rates, and the Exchange rate between the dollar that can shed some insight into the workings of the Greek economy.

Equilibrium in the Goods Market:

The equation above represents the national income equation where total output is defined as consumption as a function of income, investment as a function of the real interest rate, exports as a function the exchange rates and imports as a function of the exchange rates and income.  Government spending is exogenous and is pre-determined, but in this case it will be our variable of interest which will represent the policy variable in our system.  We are assuming that the partial derivatives of the national income equation represent the following:

  • Consumption is an increasing function of Income
  • Investment is the economy is inversely related to the Domestic Interest Rate
  • Exports are positively related to an increase in the exchange rate (depreciation of the Greek currency)
  • Imports are an increasing function of income and a decreasing function of the exchange rate (depreciation of Greek currency)

The next equation represents equilibrium in the Money market;

The equation above represents equilibrium in the money market where money demand is a function of income and domestic interest rates.  Money supply is exogenous and is determined by the European Central Bank.  We are assuming that the partial derivatives of the money demand equation follow these  basic properties:

  • Demand for money (L) is an increasing function of income
  • Domestic interest rates and the demand for money are inversely related

The next equation represent the balance of payments which asserts that net exports must be balanced by a net capital outflow and vice versa;

The equation above represents next exports or the capital account and the capital account.  The partial derivatives of the exports and imports are identical to the national income equation and the capital account has the following directional derivatives:

  • Capital will tend to flow into Greece if its’ domestic interest rate increases
  • Increases in the world interest rates will cause an outflow of capital in Greece

Writing the system described by the three equations above in implicit form we get:

In order to ensure that this system will have a solution which exist we need to assume that all partial derivatives are continuous and that the determinant of the Jacobian is non-zero.  A non-zero Jacobian rules out linear independence in the system so that there is a unique solution as demonstrated below:

Taking the differential of the system is the second step to solving this system:

This system can be written in matrix notation in the following form after dividing by the exogenous variable for the differential of government spending:

Finally, using Cramers’ we can solve for the change in Greeks gross domestic product, domestic interest rates, and the exchange rate given an increase in government expenditures:

In conclusion, as the model above shows, and increase (decrease) in Government spending by Greece will cause a temporary increase (decrease)in its gross domestic product.  This (contractionary) expansionary fiscal policy will also (decrease) increase the Greek/U.S. exchange rate (E) which essentially means (appreciating) depreciating their currency.  In Greece this mechanism has had a delayed effect, but the countrys’ reckless fiscal policy is now helping to push down the value of the Euro as speculators try and identify how to fix this problem within the Eurozone.  The fact that the European Union is not as homogeneous as it should be will cause further tough decisions to be made about their policies with countries such as Portugal, Italy, Ireland, Greece and Spain whose fiscal discipline has much to be desired.  There is not clause in the EU contract for separating a member nation and if this were to happen it would be quiet messy for all of the EU so in the mean time the bailouts will be coming for Greece, but clearly fiscal discipline must be instilled if they want to remain a respectable member of the European Union.