Given the Cobb-Douglas production function and a log utility function can be used to describe the behaviour of a given centralised representative agent economy:

y = f(l) = l,

u(c, x) = lnc + lnx

Calibration/Parameters: Productivity factor, A_G= 1; Labour elasticity, = 1/3 ; Degree to which the agent prefers leisure to work, = 0.5; Time endowment, T = 1

Further assumptions: T = l + x, where T =1

c = y = f(l,k) = A_GlK^1-, where k is constant i.e K=1 thus c = y = f(l,k) = A_Gl

^{Labour is the only factor of production since capital is constant.}

^{You may consider the lagrangian multiplier for optimization of utility and production.}

Find: the following equilibrium values for : (i)Labour (l); (ii)Consumption (c); (iii)Utility (u); (iv)Graph the equilibrium in a (c : x) space, (v)Interpret the results of the equilibrium values.

Hint: The first step should be substituting x = 1 - l; and c = y = f(l) into u(c,x) = lnc +lnx such that u(c,x) = u[ f(l) , 1- l ].

Then from here look for the first order condition (F.O.C) using the lagrangian (i.e Max lagrangian u[ f(l) , 1- l ] = ln AGl + ln (1- l).

After finding F.O.C, apply the log rule to get MPL = MRSc,x. Then solve for l , c and x. I would really appreciate an answer derived from this perspective.

PS: This is a Macroeconomics problem based on Microeconomics principles.

Thank you.