Central Bank vs. Government: Policy Games

A previous post considered various ways to model macroeconomic dynamics. This post drills down on the interaction between central banks and government fiscal policy. This interaction is much in the news these days, as the EU central bank proposes CBDCs (aka "digital prison"), and the prospect of tyranny in Canada with Mark Carney, a former central banker. Here in the US, Trump has had his dust ups with the Fed chairman Powell as well. Get ready for a rocky ride as the central bankers try to control the economies of Europe and Canada.

The interaction between a government (fiscal authority) and a central bank (monetary authority) can be modeled as a strategic game. Each player has different objectives:

• Government: Seeks to maximize economic growth and employment using fiscal policy (spending and taxation).
• Central Bank: Aims to control inflation and maintain price stability through monetary policy (interest rates and money supply).

1. Stackelberg Game Model

A Stackelberg game is a leader-follower model where one agent moves first:

• If the central bank moves first (leader), it sets interest rates based on expected government fiscal policy.
• If the government moves first, it sets spending and taxation, anticipating the central bank’s monetary response.

Mathematical Formulation

Let:

• Y_t : Output (GDP)
• G_t : Government spending
• T_t : Tax revenue
• M_t : Money supply
• i_t : Interest rate set by the central bank
• π_t : Inflation rate
• P_t : Price level

IS Curve (Goods Market Equilibrium):

Y_t = C(Y_t - T_t) + I(i_t) + G_t

Phillips Curve (Inflation Dynamics):

π_t = β (Y_t - Y^*) + π_t-1

Monetary Policy Rule (Taylor Rule Approximation):

i_t = r^* + α (π_t - π^*) + γ (Y_t - Y^*)

LM Curve (Money Market Equilibrium):

M_t - P_t = λ Y_t - η i_t

Here:

• Y^* : Potential output
• π^* : Target inflation
• r^* : Natural interest rate
• λ, η : Money demand elasticity parameters

2. Coordination Game Model

Instead of a leader-follower structure, a coordination game arises when both the central bank and government make simultaneous decisions.

Their payoffs depend on mutual policy alignment. The Nash equilibrium depends on their reaction functions:

Government’s Reaction Function:

G_t = f(i_t, Y_t, π_t, M_t)

Central Bank’s Reaction Function:

i_t = g(G_t, Y_t, π_t, M_t)

Policy Equilibrium

Solving these equations leads to an equilibrium where neither agent wants to unilaterally change its policy.

3. Implications and Outcomes

• If the central bank increases M_t, interest rates i_t decrease, stimulating investment and output Y_t, but possibly increasing inflation π_t.
• If the government overspends (G_t too high), inflation can rise, forcing the central bank to raise interest rates i_t.
• Optimal outcomes occur when both policies are coordinated, avoiding excessive inflation or recession.

This game-theoretic framework helps policymakers anticipate and mitigate conflicts between fiscal and monetary authorities.