Central Bank: The Underlying Economy 3

The source of this ambiguity becomes clear from rewriting 2.3 (see details of how to do this in the appendix’ as

Notice that the government is benevolent -in that it maximizes that portion of the workers welfare function that is unrelated to inflation- but its weight on inflation may differ from that of the workers. Of course, the case in which the government is fully benevolent, so that j3g = f3p, is just a special case of our analysis.
Solving the Game

We are now in a position to compute the equilibrium to the game among the n unions and the government. The timing of moves is as follows. Within each period £, unions move first, setting the rate of nominal wage growth ujt (г) for each worker i in each union j. The government moves next, setting the rate of price increase ttt. Given these two moves and the inherited prior individual real wages (г) and aggregate real wage Wt~i, 2.11 and 2.16 give the contemporary real wages Wt (г) and Wt. Finally, the firm sets employment and output by moving along its labor demand curve.

Notice that this timing leaves the government free to move after the unions, with no precommitment of any kind. Hence, in the jargon of monetary policy games, we are computing an equilibrium in which policy is set in а “discretionary” fashion.

We solve the stage game backwards. Since the government moves last, consider its problem first. It maximizes 2.15 subject to the technology 2.1, the firm’s optimality conditions 2.3 and 2.6, the dividend rule 2.7, and the wage laws of motion 2.11 and 2.16.