Aside from the exogenously imposed price rigidity, this model is built up from microeconomic foundations.
New Keynesian IS Curve
The primary difference from the traditional curve is the presence of future output expectations on the right side of the curve. The most important feature of this curve is that it implies an inverse relationship between the real interest rate and output. This has implications when introducing trade.
\(\ln Y_t = \ln Y_{t + 1} - \frac{1}{\theta} r_t\)
Thus we have a simple but crucial result: with nominal rigidity, monetary disturbances have real effects.
New Keynesian LM Curve
In this curve, the price level is always fixed, thus,
\(\frac{M}{P} = Y_t ^{\frac{\theta}{v}} (\frac{1 + i_t}{i_t}) ^{\frac{1}{v}}\)
\(P_t = \bar P\)
The money demand is increasing in output and decreasing in the nominal interest rate.
Mundell-Fleming Model
The Mundell-Fleming is one of the most important macro models researchers and policymakers use to analyze the short-run effects of fiscal and monetary policy. The emphasis of the model is necessarily in the short-run, as one of its main assumptions is the stickiness of prices. It is based on a traditional Keynesian model, augmented by the addition of capital mobility equations. Endogenous variables are now income, domestic interest rates, and (under floating exchange rates) the real exchange rate (so BoP is set to zero).
IS Curve
\(Y = C + I + G + X\)
\(C = C(Y - T, i - \pi ^e)\), where T is taxes, i is interest rate, and the pi term is expected inflation. Higher disposable income or lower real interest rate leads to higher consumption spending.
\(I = I(i - \pi ^e, Y_{-1})\) where the last Y term is expected income. Higher lagged income or lower real interest rate leads to higher investment spending.
\(G = \bar G\)
Thus, the solution to this component is
\(Y = C(Y - T, i - \pi ^e) + I(i - \pi ^e, Y_{-1}) + \bar G + X(\rho, Y, Y^*)\)
LM Curve
\(\frac{M}{P} = L(i, Y)\)
A higher interest rate or lower income (GDP) level leads to lower money demand.
BoP Schedule
\(BoP = CA + KA\), or current account surplus plus the capital account surplus
\(CA = X(\rho, Y, Y^*)\), where X stands for net exports, the rho term signifies the real exchange rate, the first Y term is domestic output, and the Y* is combined GDP of foreign trading partners. Higher domestic income (GDP) leads to more spending on imports and hence lower net exports. Higher foreign income leads to higher spending by foreigners on the country's exports and thus higher next exports. A higher nominal exchange rate leads to lower net exports.
\(\rho \equiv \frac{eP^*}{P}\), where e is nominal exchange rate and P and P* are domestic and international prices respectively
\(e \equiv \frac{DC}{FC}\), where the amount of domestic currency is divided by foreign currency
\(KA = \sigma ( i - i^*) + k\), where sigma is the sensitivity to interest rates, i is domestic interest, and i* is the foreign interest rate. The derivative of the function of sigma is the degree of capital mobility (effect of differences between domestic and foreign interest rates upon capital flows, KA).
Thus, the solution to this component is
\(BoP = X(\rho, Y, Y^*) + \sigma ( i - i^*) + k\)
Using this information, we can illustrate the model given perfect capital mobility or no capital mobility:
Fiscal Multiplier
Fiscal policy is the most powerful when an exchange rate is fixed, rather than floating. Using Cramer's rule, we can determine the impact of increased government expenditure on output:
\((\frac{dy}{d \bar G}) = ( \frac{L_i}{(1 - c_y - nx_y) L_i + (c_r + I_r)L_y - (L_y \sigma - nx_y L_i)} ) > 0\)
Lucas Island Model
The nominal imperfection we have focused on so far is the cost of changing nominal prices. Long before work on menu costs, Lucas and Phelps suggested a difference nominal imperfection: perhaps producers do not observe the aggregate price level perfectly. If a producer does not know the price level, then it does not know whether a change in the price level reflects a change in the good's relative price or a change in the aggregate price.
Lucas assumed rational expectations, which at the time was controversial but has since come to be a stable in economic theory. In the short-run, there is a Phillips curve relationship between inflation and unemployment (as inflation is positively related with output and negatively with unemployment), however, in the long-run the producer isn't tricked by anticipated inflation. This results in policy ineffectiveness since inflation cannot induce increases to output.
The key implications of the model are that the increase in the money supply raises aggregate demand, and thus produces an outward shift in the demand curve for each good. Each supplier's best guess is that some portion of the rise in demand reflects a relative price shock. Thus producers increase their output. In the long-run, if the producer anticipates inflation he will not be induced to increase output, though unanticipated changes in monetary policy will cause him to react.