This lecture continues our analysis in this lecture

{doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` about the model that Tjalling Koopmans {cite}`Koopmans`

and David Cass {cite}`Cass` used to study optimal growth.

and David Cass {cite}`Cass` used to study optimal capital accumulation.

This lecture illustrates what is, in fact, a

more general connection between a **planned economy** and an economy

organized as a **competitive equilibrium**.

organized as a competitive equilibrium or a **market economy**.

The earlier lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` studied a planning problem and used ideas including

- A min-max problem for solving the planning problem.

- A Lagrangian formulation of the planning problem that leads to a system of difference equations.

- A **shooting algorithm** for solving difference equations subject

to initial and terminal conditions.

- A **turnpike** property that describes optimal paths for

long-but-finite horizon economies.

The present lecture uses additional ideas including

- Hicks-Arrow prices named after John R. Hicks and Kenneth Arrow.

- A connection between some Lagrange multipliers in the min-max

- Hicks-Arrow prices, named after John R. Hicks and Kenneth Arrow.

- A connection between some Lagrange multipliers from the planning

problem and the Hicks-Arrow prices.

- A **Big** $K$ **, little** $k$ trick widely used in

macroeconomic dynamics.

* We shall encounter this trick in [this lecture](https://python.quantecon.org/rational_expectations.html)

and also in [this lecture](https://python-advanced.quantecon.org/dyn_stack.html).

- A non-stochastic version of a theory of the **term structure of

interest rates**.

- An intimate connection between the cases for the optimality of two

competing visions of good ways to organize an economy, namely:

- An intimate connection between two

ways to organize an economy, namely:

* **socialism** in which a central planner commands the

allocation of resources, and

* **capitalism** (also known as **a market economy**) in

* **competitive markets** in

which competitive equilibrium **prices** induce individual

consumers and producers to choose a socially optimal allocation

as an unintended consequence of their selfish

as unintended consequences of their selfish

decisions

Let's start with some standard imports:

In this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>`, we studied a problem in which a planner chooses an allocation $\{\vec{C},\vec{K}\}$ to

maximize {eq}`utility-functional` subject to {eq}`allocation`.

The allocation that solves the planning problem plays an important role in a competitive equilibrium as we shall see below.

The allocation that solves the planning problem reappears in a competitive equilibrium, as we shall see below.

## Competitive Equilibrium

But now there is no planner.

Market prices adjust to reconcile distinct decisions that are made

There are (unit masses of) price taking consumers and firms.

Market prices are set to reconcile distinct decisions that are made

separately by a representative household and a representative firm.

There is a representative consumer who has the same preferences over

consumption plans as did the consumer in the planned economy.

consumption plans as did a consumer in the planned economy.

Instead of being told what to consume and save by a planner, the

household chooses for itself subject to a budget constraint

Instead of being told what to consume and save by a planner, a

consumer (also known as a *household*) chooses for itself subject to a budget constraint

- At each time $t$, the household receives wages and rentals

of capital from a firm -- these comprise its **income** at

- The representative household and the representative firm are both

**price takers** who believe that prices are not affected by their choices

**Note:** We can think of there being a large number

$M$ of identical representative consumers and $M$

**Note:** We can think of there being unit measures of identical representative consumers and

identical representative firms.

## Market Structure

There are sequences of prices

$\{w_t,\eta_t\}_{t=0}^T= \{\vec{w}, \vec{\eta} \}$

where $w_t$ is a wage or rental rate for labor at time $t$ and

$\eta_t$ is a rental rate for capital at time $t$.

where

- $w_t$ is a wage or rental rate for labor at time $t$

In addition there is are intertemporal prices that work as follows.

- $\eta_t$ is a rental rate for capital at time $t$

Let $q^0_t$ be the price of a good at date $t$ relative

In addition there is a vector $\{q_t^0\}$ of intertemporal prices where

- $q^0_t$ is the price of a good at date $t$ relative

to a good at date $0$.

We call $\{q^0_t\}_{t=0}^T$ a vector of **Hicks-Arrow prices**,

named after the 1972 economics Nobel prize winners.

Evidently,

Units of $q_t^0$ could be

$$

q^0_t=\frac{\text{number of time 0 goods}}{\text{number of time t goods}}

\frac{\text{number of time 0 goods}}{\text{number of time t goods}}

$$

Because $q^0_t$ is a **relative price**, the units in terms of

which prices are quoted are arbitrary -- we are free to normalize them.

But because $q^0_t$ is a **relative price**, the units in terms of

which prices are quoted are arbitrary, we are free to re-normalize them.

## Firm Problem

If $\frac{\partial F}{\partial \tilde k_t}> \eta_t$, then the

firm makes positive profits on each additional unit of

$\tilde k_t$, so it will want to make $\tilde k_t$

$\tilde k_t$, so it would want to make $\tilde k_t$

arbitrarily large.

But setting $\tilde k_t = + \infty$ is not physically feasible,

$\frac{\partial F}{\partial \tilde n_t}> w_t$.

If $\frac{\partial \tilde k_t}{\partial \tilde k_t}< \eta_t$,

the firm will set $\tilde k_t$ to zero, something that is not feasible.

the firm would want to set $\tilde k_t$ to zero, which is not feasible.

It is convenient to define

$\vec{w} =\{w_0, \dots,w_T\}$and $\vec{\eta}= \{\eta_0, \dots, \eta_T\}$.

k_{T+1}: \quad -\lambda q_0^{T+1} \leq 0, \ \leq 0 \text{ if } k_{T+1}=0; \ =0 \text{ if } k_{T+1}>0

```

Now we plug in our guesses of prices and embark on some algebra in the hope of derived all first-order necessary conditions

Now we plug in our guesses of prices and embark on some algebra in the hope of recovering all first-order necessary conditions

{eq}`constraint1`-{eq}`constraint4` for the planning problem from this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>`.

Combining {eq}`cond1` and {eq}`eq-price`, we get:

\sum_{t=0}^T \beta^t \mu_{t} \left(C_t+ (K_{t+1} -(1-\delta)K_t)-f(K_t)+K_t f'(K_t)-f'(K_t)K_t\right) \leq 0

$$

which simplifies

which simplifies to

$$

\sum_{t=0}^T \beta^t \mu_{t} \left(C_t +K_{t+1} -(1-\delta)K_t - F(K_t,1)\right) \leq 0

which is exactly {eq}`eq-pr4`.

So at our guess for the equilibrium price system, the allocation

Thus, at our guess for the equilibrium price system, the allocation

that solves the planning problem also solves the problem faced by a firm

within a competitive equilibrium.

plot_yield_curves(pp, 20, 0.3, k_ss/3, T_arr)

```

We aim to have more to say about the term structure of interest rates

in a planned lecture on the topic.