Question

2 Calculating a Pareto optimal allocation Suppose the representative household has the following utility function: U (C,) InC +0.5ln l where C is consumption and 1 is leisure. The household's time constraint is I+N-1 where Ns is the representative household's labour supply. Further assume that the production function is Cobb-Douglas 0.5 0.5 where 2-1 and K = 1 2.1 Assuming that the government spending is G = 0, use the Social Planners problem to solve for Pareto optimal numerical values C.r.N*.Y (8 points) 2.2 Find the numerical values of the real wage rate and of the representa- tive firm's profit π* in a competitive equilibrium implementing the Social Planner's (Pareto) optimal solution (4 points) 2.3 Draw a carefully labeled chart showing the impact of an increase in G on the Pareto Optimal allocation. Your chart does not have to be drawn exactly to scale. (7 points) 2.4 How would the optimal values C *,N". Y*,w*, π* dange (in what di- rection) in case if government spending increases (6 points)?

Answer 1

The social planner has to maximise the consumer's utility. So the social planner solves:

max   $u(C,N)$
subject to. $C = f(l)$
and   $l + N = 1$

In order to solve the planning problem, we use Lagrange multipliers. Let $\phi_{t}$ denote the Lagrange multiplier on the time constraint and let $\phi_{c}$ denote the Lagrange multiplier on the consumption constraint.

Since there are two constraints, the Lagrangean for this problem is given by:
$L(C,N,l,\phi_{c},\phi_{t}) = u(C,N) + \phi_{c}(f(l) - c) + \phi_{t}(1 - l - N)$

For our problem, the equations are given as:
max $ln(C) + 0.5 ln(l)$ = $ln(C) + 0.5ln(1-N)$
subject to. $C = (N)^{1/2}$ = $(1-l)^{1/2}$
  $l + N = 1$

The first order conditions for this problem are given by:
[C] : $\frac{\partial u}{\partial C}$ = $\phi_{c}^{*}$ = 1/C
[N] : $\frac{\partial u}{\partial N}$ = $\phi_{t}^{*}$ = $\frac{-0.5}{1-N^{*}}$

[l] :  $\phi_{c}^{*}$$\times (1/2)(1 - l^{*})^{-1/2} = \phi_{t}^{*}$
[$\phi_{c}$] : $C^{*} = (1 - l^{*})^{1/2}$
[$\phi_{t}$] : $l^{*} + N^{*} = 1$

The superscript * indicates that the quantities are the optimal quantities of the planning problem.
Thus, solving, we get optimal quantities as
$l^{*} = 1/2$
$N^{*} = 1/2$
$C^{*} = (1/2)^{0.5}$
The value of  $\phi_{t}^{*}$/ $\phi_{c}^{*}$ gives us a value precisely equal to the equilibrium real wage.

When government spending exists, the equation of C = f(l) gets an additional factor, and becomes
$C = f(l) - G$
It affects the equilibrium optimal conditions as shown in the graph (the value of C decreases by Gz and the value of l decreases by (G/z):