Hello. I have problems solving this exercise. If you could help me to solve this microfoundation problem, I would be very grateful!

Consider the following closed economy. There is a representative household who cares about consumption of a homogeneous good and leisure. His preferences can be summarized with the following utility function:

U = ln(c) + ln(h) + 1 /2 ln(G)

l + h = 1

c = wl T

The household's problem is to choose the allocation (c, h, l) that maximizes utility subject to the constraints, taking as given (G, T, w).

On the production side, there is a representative competitive firm that hires labor to produce the final good, using a linear technology:

Y = L

Goal is to maximaze profits: max Y wL (with respect to L)

G = T

The market-clearing conditions for this economy are as follows. The labor market clears:

l = L and the good market clears:

c + G = Y

Questions:

1. Define an equilibrium for this economy.

2. Solve for the equilibrium using G = 1/4, and compute the output and consumption fiscal multipliers.