Macroeconomics 3

Neoclassical Growth Model

Krueger - Macroeconomics Theory

The neoclassical growth model is arguably the single most important workhorse in modern macroeconomics. It is widely used in growth theory, business cycle theory and quantitative applications in public finance.

In each period there are three goods that are traded, labor services $n_t$; capital services $k_t$ and a final output good $y_t$ that can be either consumed, $c_t$ or invested, $i_t$. As usual for a complete description of the economy we have to specify technology, preferences, endow- ments and the information structure.

1. Technology: $y_{t}=F\left(k_{t}, n_{t}\right)$, $y_{t}=i_{t}+c_{t}$, $k_{t+1}=(1-\delta) k_{t}+i_{t}$,
2. Preferences: $u\left(\left\{c_{t}\right\}_{t=0}^{\infty}\right)=\sum_{t=0}^{\infty} \beta^{t} U\left(c_{t}\right)$
3. Endowments: At period 0 each household is born with endowments $\bar{k}_0$ of initial capital. Furthermore each household is endowed with one unit of productive time in each period, to be devoted either to leisure or to work.
4. Information households and firms have perfect foresight.
5. Equilibrium

Optimal Growth: Pareto Optimal

The problem of the planner is

$$\begin{aligned} w\left(\bar{k}_{0}\right) &=\max _{\left\{c_{t}, k_{t}, n_{t}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} U\left(c_{t}\right) \\ \text { s.t. } \quad F\left(k_{t}, n_{t}\right) &=c_{t}+k_{t+1}-(1-\delta) k_{t} \\ c_{t} & \geq 0, k_{t} \geq 0,0 \leq n_{t} \leq 1 \\ k_{0} & \leq \bar{k}_{0} \end{aligned}$$

Define since no leisure in utility, $n_t=1$,

$f(k) = F(k,1)+(1-\delta)k$

rewrite the social plannerís problem as

$$\begin{aligned} w\left(\bar{k}_{0}\right) &=\max _{\left\{k_{t+1}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} U\left(f\left(k_{t}\right)-k_{t+1}\right) \\ 0 & \leq k_{t+1} \leq f\left(k_{t}\right) \\ k_{0} &=\bar{k}_{0}>0 \text { given } \end{aligned}$$

Recursive Formulation

$$\begin{aligned} w\left(k_{0}\right)=& \max _{\substack{\left\{k_{t+1}\right\}_{t=0}^{\infty} \\ 0 \leq k_{t+1} \leq f\left(k_{t}\right), k_{0} \text { given }}} \sum_{t=0}^{\infty} \beta^{t} U\left(f\left(k_{t}\right)-k_{t+1}\right) \\ =& \max _{\substack{k_{1} \text { s.t. } \\ 0 \leq k_{1} \leq f\left(k_{0}\right), k_{0} \text { given }}}\left\{U\left(f\left(k_{0}\right)-k_{1}\right)+\beta\left[\max _{\substack{\left\{k_{t+1}\right\}_{t=1}^{\infty} \\ 0 \leq k_{t+1} \leq f\left(k_{t}\right), k_{1} \text { given }}} \sum_{t=1}^{\infty} \beta^{t-1} U\left(f\left(k_{t}\right)-k_{t+1}\right)\right\}\right\} \end{aligned}$$

Bellman equiation This function $v$ (the so-called value function) solves the following recursion,

$$v(k)=\max _{0 \leq k^{\prime} \leq f(k)}\left\{U\left(f(k)-k^{\prime}\right)+\beta v\left(k^{\prime}\right)\right\}$$

tip

Note again that v and w are two very di§erent functions; v is the value function for the recursive formulation of the planners problem and w is the corresponding function for the sequential problem. we want to establish that v = w.

By solving the functional equation we mean finding a value function $v$ solving and an optimal ==policy function== $k^{'}= g(k)$ that describes the optimal $k^{'}$ from the maximization part as a function of $k$ i.e. for each possible value that k can take.

• guess
• Analytical Approach
• Numerical Approach

Euler Equation Approach and Transversality Conditions

Note that this approach also, as the guess and verify method, will only work in very simple examples, but not in general, whereas the recursive numerical approach works for a wide range of parameterizations of the neoclassical growth model.

$$L=U\left(f\left(k_{0}\right)-k_{1}\right)+\ldots+\beta^{t} U\left(f\left(k_{t}\right)-k_{t+1}\right)+\beta^{t+1} U\left(f\left(k_{t+1}\right)-k_{t+2}\right)+\ldots+\beta^{T} U\left(f\left(k_{T}\right)-k_{T+1}\right)$$

$$\frac{\partial L}{\partial k_{t+1}}=-\beta^{t} U^{\prime}\left(f\left(k_{t}\right)-k_{t+1}\right)+\beta^{t+1} U^{\prime}\left(f\left(k_{t+1}\right)-k_{t+2}\right) f^{\prime}\left(k_{t+1}\right)=0$$

tip

This first order condition some times is called an Euler equation (supposedly because it is loosely linked to optimality conditions in continuous time calculus of variations, developed by Euler)

The Infinite Horizon Case

Again the eular eqution is a second order difference equation, but now we only have an initial condition for $k_0$ but no terminal condition since there is no terminal time period.

In a lot of applications, the ==transversality condition== substitutes for the missing terminal condition.

tip

The transversality condition states that the value of the capital stock $k_t$; when measured in terms of discounted utility, goes to zero as time goes to infinity.

Steady States and the Modified Golden Rule

A steady state is defined as a social optimum or competitive equilibrium in which allocations are constant over time, $c_t=c^{*}, k_t=k^{*}$

The Euler equations for the social planner problem read as

$$\beta U^{\prime}\left(c_{t+1}\right) f^{\prime}\left(k_{t+1}\right)=U^{\prime}\left(c_{t}\right)$$

In a steady state

$$\begin{aligned} &\beta f^{\prime}\left(k^{*}\right) =1 \\ &f^{\prime}\left(k^{*}\right) =1+\rho \end{aligned}$$

Given $f^{\prime}(k)=F_{k}(k, 1)+1-\delta$, we obtain the so-called ==modified golden rule== $F_{k}\left(k^{*}, 1\right)-\delta=\rho$

tip

that is, the social planner sets the marginal product of capital, net of depreciation, equal to the time discount rate. The capital stock that maximizes consumption per capita, called the (original) golden rule.

As we will see below, the net ==real interest rate== in a competitive equilibrium equals $F_{k}\left(k, 1\right)-\delta$ (euler equation). So the modified golden rule can be restated as equating the real interest rate and the time discount rate.

Competitive Equilibrium Growth

What we are genuinely interested in are allocations and prices that arise when firms and consumers interact in markets. In this section we will discuss the connection between Pareto optimal allocations and allocations arising in a competitive equilibrium.

• we assume that consumers own all factors of production and rent it out to the firms.
• households own the firms, i.e. are claimants of the firms' profits.
• final goods market and the factor markets are perfectly competitive.
• a single market at time zero in which goods for all future periods are traded. After this market closes, in all future periods the agents in the economy just carry out the trades they agreed upon in period 0. (==Arrow-Debreu Market==)

For each period there are three goods that are traded:

1. The final output good $y_t=c_t+i_t$, Let $p_t$ denote the ==price== of the period t final output good, quoted in period 0. We let the period 0 output good be the ==numeraire== and thus normalize $p_0 = 1$.
2. Labor services $n_t$. Let $w_t$ be the price of one unit of labor services delivered in period t, quoted in period 0, in terms of the period t consumption good. Hence $w_t$ is the ==real wage==, it tells how many units of the period t consumption goods one can buy for the receipts for one unit of labor. The wage in terms of the numeraire, the period 0 output good is $p_t w_t$
3. Capital services $k_t$. Let $r_t$ be the rental price of one unit of capital services delivered in period t, quoted in period 0, in terms of the period t consumption good. Note that rt is the real rental rate of capital, the rental rate in terms of the numeraire good is $p_t r_t$ .

Firm

$$\begin{aligned} & \pi =\max _{\left\{y_{t}, k_{t}, n_{t}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} p_{t}\left(y_{t}-r_{t} k_{t}-w_{t} n_{t}\right) \\ \text { s.t. } \quad & y_{t} =F\left(k_{t}, n_{t}\right) \text { for all } t \geq 0 \\ & y_{t}, k_{t}, n_{t} \geq 0 \end{aligned}$$

FOC

$$\begin{aligned} r_{t} &=F_{k}\left(k_{t}, n_{t}\right) \\ w_{t} &=F_{n}\left(k_{t}, n_{t}\right) \end{aligned}$$

gives $\pi_{t}=p_{t}\left(F\left(k_{t}, n_{t}\right)-F_{k}\left(k_{t}, n_{t}\right) k_{t}-F_{n}\left(k_{t}, n_{t}\right) n_{t}\right)=0$

consumer

$k_t=x_t$,

$$\begin{aligned} &\max _{\left\{c_{t}, i_{t}, x_{t+1}, k_{t}, n_{t}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} U\left(c_{t}\right) \\ \text { s.t. } &\quad \sum_{t=0}^{\infty} p_{t}\left(c_{t}+i_{t}\right) \leq \sum_{t=0}^{\infty} p_{t}\left(r_{t} k_{t}+w_{t} n_{t}\right)+\pi \\ & x_{t+1} =(1-\delta) x_{t}+i_{t} \quad \text { all } t \geq 0 \\ & 0 \leq n_{t} \leq 1,0 \leq k_{t} \leq x_{t} \quad \text { all } t \geq 0 \\ & c_{t}, x_{t+1} \geq 0 \quad 0 \text { all } t \geq 0 \\ & x_{0} \text { given } \end{aligned}$$

FOC

$$\begin{aligned} \beta^{t} U^{\prime}\left(c_{t}\right) &=\mu p_{t} \\ \beta^{t+1} U^{\prime}\left(c_{t+1}\right) &=\mu p_{t+1} \\ \mu p_{t} &=\mu\left(1-\delta+r_{t+1}\right) p_{t+1} \end{aligned}$$

gives euler equiation:

$$\frac{\beta U^{\prime}\left(c_{t+1}\right)}{U^{\prime}\left(c_{t}\right)}=\frac{p_{t+1}}{p_{t}}=\frac{1}{1+r_{t+1}-\delta}$$

net interest rate $r_{t+1}-\delta$ with marginal pricing condition (firm FOC) :

$$r_{t}=F_{k}\left(k_{t}, 1\right)=f^{\prime}\left(k_{t}\right)-(1-\delta)$$

market clear

$$\begin{aligned} y_{t} &=c_{t}+i_{t} \text { (Goods Market) } \\ n_{t}^{d} &=n_{t}^{s} \text { (Labor Market) } \\ k_{t}^{d} &=k_{t}^{s} \text { (Capital Services Market) } \end{aligned}$$

Sequential markets equilibrium

In a sequential markets equilibrium households (who own the capital stock) take sequences of wages and interest rates as given and in every period chooses consumption and capital to be brought into tomorrow.

consumer

$$\begin{aligned} & \max _{\left\{c_{t}, k_{t+1}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} U\left(c_{t}\right) \\ & \text{s.t.} \\ & c_{t}+k_{t+1}-(1-\delta) k_{t}= w_{t}+r_{t} k_{t} \\ & c_{t}, k_{t+1} \geq 0 \\ & k_{0} \text { given } \end{aligned}$$

firm

$$\max _{k_{t}, n_{t} \geq 0} F\left(k_{t}, n_{t}\right)-w_{t} n_{t}-r_{t} k_{t}$$