closed market economy, suppose government borrows only at one period duration. Let $B_t$ be the number of apples government borrowed at time $t$, which amounts to $B_t(1+r_t)$ apples payment at time $t+1$ with the interest. Needless to say, government can borrow at time $t+1$ as well to meet this payment. Note that $B_t$ is allowed to be positive which means government is lending rather than borrowing in the financial market. Let $G_t$ and $T_t$ be the government spending and taxes collected, then the government budget constraint at time $t$ is simply
$G_t + B_{t-1}(1+r_{t-1}) = B_t + T_t$
(0.3)
where the left-hand side $G_t + B_{t-1}(1+r_{t-1})$ is the total outflow of apples from the government budget at time $t$ for spending and debt obligations, and the right-hand side $B_t + T_t$ is the total inflow of apples to the government budget from taxes and new borrowings. The government budget constraint (Equation 0.3) must hold at any period $t$. For simplicity, assume that government taxes the income at rate $\kappa$ which is the only revenue of the government. Individuals consume their after-tax income at rate $MPC$. Suppose the economy is in a steady-state initially where government spending is fixed at $G$. Derive the levels of capital stock, real interest rate and government debt stock in this steady state. Suppose government spending permanently increased at time $t^*$ from $G$ to $G'$, draw evolutions of capital stock, real interest rate and government debt stock over time. Assume that technology and labor supply don't change over time.
