Consider a standard Arrow$-$Debreu model in which the representative firm is the

owner of capital. The consumption good is produced by means of a constant returns

to scale technology:

$Y_t=F(K_t,1)=f(K_t)$

The representative firm maximizes the present value of profits:

${\displaystyle \underset{t=0}{\overset{}{}}}{p}_t[Y_t-I_t-{}_t]$

where $p_t$ is the Arrow$-$Debreu price at $t,$ and $I_t$ denotes investment in physical capital

$($the firm accumulates capital$)$ subject to the technological constraint and the usual

law of motion for capital. Consumers do not directly own physical capital, but are

the owners of the representative firm and, hence, each period receive the profits gener$-$

ated by the company $($note that even though technology exhibits constant returns to

scale, profits are not necessarily zero in this economy$).$ The representative consumer

maximizes its utility subject to the budget constraint:

${\displaystyle \underset{t=0}{\overset{}{}}}{p}_t{c}_t={\displaystyle \underset{t=0}{\overset{}{}}}{p}_t[{}_t+d_t]$

where $d_t$ denotes profits generated by the firm at date $t.$ Assume that the consumer

$3$

has access to a one$-$period bond. These bonds are issued and negotiated between

households $($its aggregate net supply is zero$).$

$($a$)$ Define an Arrow Debreu Competitive Equilibrium for this economy and

find the conditions characterizing it$.$ What is the equilibrium price of each bond?

$($b$)$ Define the Planner's Problem and provide the conditions characterizing its

solution. Are the welfare theorems satisfied? Is the allocation of resources in the

Sequential Equilibrium equivalent to the one in which physical capital is owned

by the representative consumer?

Consider now a sequential economy with the same characteristics: the firm is the

owner of physical capital, etc. The consumer now can negotiate bonds issued by

households $($its aggregate net supply is zero$).$

$($c$)$ Assume that consumers are the owners of the representative company by means

of stocks negotiated at each $t.$ Show that in equilibrium the total market value