diff --git a/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png b/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png index bb1ace2a..c36dce06 100644 Binary files a/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png and b/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png differ diff --git a/lectures/cons_smooth.md b/lectures/cons_smooth.md index bc43a5d4..886ed823 100644 --- a/lectures/cons_smooth.md +++ b/lectures/cons_smooth.md @@ -17,17 +17,28 @@ kernelspec: ## Overview -Technically, this lecture is a sequel to this quantecon lecture {doc}`present values `, although it might not seem so at first. -It will take a while for a "present value" or asset price explicilty to appear in this lecture, but when it does it will be a key actor. +In this lecture, we'll study a famous model of the "consumption function" that Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`) proposed to fit some empirical data patterns that the original Keynesian consumption function described in this quantecon lecture {doc}`geometric series ` missed. + +In this lecture, we'll study what is often called the "consumption-smoothing model" using matrix multiplication and matrix inversion, the same tools that we used in this quantecon lecture {doc}`present values `. + +Formulas presented in {doc}`present value formulas` are at the core of the consumption smoothing model because we shall use them to define a consumer's "human wealth". -In this lecture, we'll study a famous model of the "consumption function" that Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`) proposed to fit some empirical data patterns that the simple Keynesian model described in this quantecon lecture {doc}`geometric series ` had missed. +The key idea that inspired Milton Friedman was that a person's non-financial income, i.e., his or +her wages from working, could be viewed as a dividend stream from that person's ``human capital'' +and that standard asset-pricing formulas could be applied to compute a person's +``non-financial wealth'' that capitalizes the earnings stream. + +```{note} +As we'll see in this quantecon lecture {doc}`equalizing difference model `, +Milton Friedman had used this idea in his PhD thesis at Columbia University, +eventually published as {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`. +``` -The key insight of Friedman and Hall was that today's consumption ought not to depend just on today's non-financial income: it should also depend on a person's anticipations of her **future** non-financial incomes at various dates. +It will take a while for a "present value" or asset price explicilty to appear in this lecture, but when it does it will be a key actor. -In this lecture, we'll study what is sometimes called the "consumption-smoothing model" using only linear algebra, in particular matrix multiplication and matrix inversion. -Formulas presented in {doc}`present value formulas` are at the core of the consumption smoothing model because they are used to define a consumer's "human wealth". +## Analysis As usual, we'll start with by importing some Python modules. @@ -39,13 +50,13 @@ from collections import namedtuple +++ {"user_expressions": []} -Our model describes the behavior of a consumer who lives from time $t=0, 1, \ldots, T$, receives a stream $\{y_t\}_{t=0}^T$ of non-financial income and chooses a consumption stream $\{c_t\}_{t=0}^T$. +The model describes a consumer who lives from time $t=0, 1, \ldots, T$, receives a stream $\{y_t\}_{t=0}^T$ of non-financial income and chooses a consumption stream $\{c_t\}_{t=0}^T$. We usually think of the non-financial income stream as coming from the person's salary from supplying labor. -The model takes that non-financial income stream as an input, regarding it as "exogenous" in the sense of not being determined by the model. +The model takes a non-financial income stream as an input, regarding it as "exogenous" in the sense of not being determined by the model. -The consumer faces a gross interest rate of $R >1$ that is constant over time, at which she is free to borrow or lend, up to some limits that we'll describe below. +The consumer faces a gross interest rate of $R >1$ that is constant over time, at which she is free to borrow or lend, up to limits that we'll describe below. To set up the model, let @@ -65,22 +76,27 @@ To set up the model, let * $a_{T+1} \geq 0$ be a terminal condition on final assets -While the sequence of financial wealth $a$ is to be determined by the model, it must satisfy two **boundary conditions** that require it to be equal to $a_0$ at time $0$ and $a_{T+1}$ at time $T+1$. +The sequence of financial wealth $a$ is to be determined by the model. + +We require it to satisfy two **boundary conditions**: + + * it must equal an exogenous value $a_0$ at time $0$ + * it must equal or exceed an exogenous value $a_{T+1}$ at time $T+1$. -The **terminal condition** $a_{T+1} \geq 0$ requires that the consumer not die leaving debts. +The **terminal condition** $a_{T+1} \geq 0$ requires that the consumer not leave the model in debt. -(We'll see that a utility maximizing consumer won't **want** to die leaving positive assets, so she'll arrange her affairs to make -$a_{T+1} = 0.) +(We'll soon see that a utility maximizing consumer won't **want** to die leaving positive assets, so she'll arrange her affairs to make +$a_{T+1} = 0$.) -The consumer faces a sequence of budget constraints that constrains the triple of sequences $y, c, a$ +The consumer faces a sequence of budget constraints that constrains sequences $(y, c, a)$ $$ a_{t+1} = R (a_t+ y_t - c_t), \quad t =0, 1, \ldots T $$ (eq:a_t) -Notice that there are $T+1$ such budget constraints, one for each $t=0, 1, \ldots, T$. +Equations {eq}`eq:a_t` constitute $T+1$ such budget constraints, one for each $t=0, 1, \ldots, T$. -Given a sequence $y$ of non-financial income, there is a big set of **pairs** $(a, c)$ of (financial wealth, consumption) sequences that satisfy the sequence of budget constraints {eq}`eq:a_t`. +Given a sequence $y$ of non-financial incomes, a large set of **pairs** $(a, c)$ of (financial wealth, consumption) sequences satisfy the sequence of budget constraints {eq}`eq:a_t`. Our model has the following logical flow. @@ -93,19 +109,19 @@ Our model has the following logical flow. * If it does, declare that the candidate path is **budget feasible**. - * if the candidate consumption path is not budget feasible, propose a path with less consumption sometimes and start over + * if the candidate consumption path is not budget feasible, propose a less greedy consumption path and start over Below, we'll describe how to execute these steps using linear algebra -- matrix inversion and multiplication. The above procedure seems like a sensible way to find "budget-feasible" consumption paths $c$, i.e., paths that are consistent with the exogenous non-financial income stream $y$, the initial financial asset level $a_0$, and the terminal asset level $a_{T+1}$. -In general, there will be many budget feasible consumption paths $c$. +In general, there are **many** budget feasible consumption paths $c$. -Among all budget-feasible consumption paths, which one **should** the consumer want to choose? +Among all budget-feasible consumption paths, which one **should** a consumer want? -We shall eventually evaluate alternative budget feasible consumption paths $c$ using the following **welfare criterion** +To answer this question, we shall eventually evaluate alternative budget feasible consumption paths $c$ using the following utility functional or **welfare criterion**: ```{math} :label: welfare @@ -115,9 +131,9 @@ W = \sum_{t=0}^T \beta^t (g_1 c_t - \frac{g_2}{2} c_t^2 ) where $g_1 > 0, g_2 > 0$. -The fact that the utility function $g_1 c_t - \frac{g_2}{2} c_t^2$ has diminishing marginal utility imparts a preference for consumption that is very smooth when $\beta R \approx 1$. +When $\beta R \approx 1$, the fact that the utility function $g_1 c_t - \frac{g_2}{2} c_t^2$ has diminishing marginal utility imparts a preference for consumption that is very smooth. -Indeed, we shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`), this criterion assigns higher welfare to **smoother** consumption paths. +Indeed, we shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`), criterion {eq}`welfare` assigns higher welfare to **smoother** consumption paths. By **smoother** we mean as close as possible to being constant over time. @@ -127,7 +143,7 @@ Let's dive in and do some calculations that will help us understand how the mode Here we use default parameters $R = 1.05$, $g_1 = 1$, $g_2 = 1/2$, and $T = 65$. -We create a namedtuple to store these parameters with default values. +We create a Python **namedtuple** to store these parameters with default values. ```{code-cell} ipython3 ConsumptionSmoothing = namedtuple("ConsumptionSmoothing", @@ -145,7 +161,7 @@ def creat_cs_model(R=1.05, g1=1, g2=1/2, T=65): ## Friedman-Hall consumption-smoothing model -A key object in the model is what Milton Friedman called "human" or "non-financial" wealth at time $0$: +A key object is what Milton Friedman called "human" or "non-financial" wealth at time $0$: $$ @@ -153,9 +169,9 @@ h_0 \equiv \sum_{t=0}^T R^{-t} y_t = \begin{bmatrix} 1 & R^{-1} & \cdots & R^{-T \begin{bmatrix} y_0 \cr y_1 \cr \vdots \cr y_T \end{bmatrix} $$ -Human or non-financial wealth is evidently just the present value at time $0$ of the consumer's non-financial income stream $y$. +Human or non-financial wealth at time $0$ is evidently just the present value of the consumer's non-financial income stream $y$. -Notice that formally it very much resembles the asset price that we computed in this quantecon lecture {doc}`present values `. +Formally it very much resembles the asset price that we computed in this quantecon lecture {doc}`present values `. Indeed, this is why Milton Friedman called it "human capital". @@ -165,22 +181,22 @@ $$ a_{T+1} = 0, $$ -it is possible to convert a sequence of budget constraints into the single intertemporal constraint +it is possible to convert a sequence of budget constraints {eq}`eq:a_t` into a single intertemporal constraint -$$ -\sum_{t=0}^T R^{-t} c_t = a_0 + h_0, -$$ +$$ +\sum_{t=0}^T R^{-t} c_t = a_0 + h_0. +$$ (eq:budget_intertemp) -which says that the present value of the consumption stream equals the sum of finanical and non-financial (or human) wealth. +Equation {eq}`eq:budget_intertemp` says that the present value of the consumption stream equals the sum of finanical and non-financial (or human) wealth. -Robert Hall {cite}`Hall1978` showed that when $\beta R = 1$, a condition Milton Friedman had also assumed, -it is "optimal" for a consumer to **smooth consumption** by setting +Robert Hall {cite}`Hall1978` showed that when $\beta R = 1$, a condition Milton Friedman had also assumed, it is "optimal" for a consumer to **smooth consumption** by setting $$ c_t = c_0 \quad t =0, 1, \ldots, T $$ -(Later we'll present a "variational argument" that shows that this constant path is indeed optimal when $\beta R =1$.) +(Later we'll present a "variational argument" that shows that this constant path maximizes +criterion {eq}`welfare` when $\beta R =1$.) In this case, we can use the intertemporal budget constraint to write @@ -194,23 +210,22 @@ Equation {eq}`eq:conssmoothing` is the consumption-smoothing model in a nutshell ## Mechanics of Consumption smoothing model -As promised, we'll provide step by step instructions on how to use linear algebra, readily implemented -in Python, to compute all the objects in play in the consumption-smoothing model. +As promised, we'll provide step-by-step instructions on how to use linear algebra, readily implemented in Python, to compute all objects in play in the consumption-smoothing model. In the calculations below, we'll set default values of $R > 1$, e.g., $R = 1.05$, and $\beta = R^{-1}$. ### Step 1 -For some $(T+1) \times 1$ $y$ vector, use matrix algebra to compute $h_0$ +For a $(T+1) \times 1$ vector $y$, use matrix algebra to compute $h_0$ $$ -\sum_{t=0}^T R^{-t} y_t = \begin{bmatrix} 1 & R^{-1} & \cdots & R^{-T} \end{bmatrix} +h_0 = \sum_{t=0}^T R^{-t} y_t = \begin{bmatrix} 1 & R^{-1} & \cdots & R^{-T} \end{bmatrix} \begin{bmatrix} y_0 \cr y_1 \cr \vdots \cr y_T \end{bmatrix} $$ ### Step 2 -Compute the optimal level of consumption $c_0 $ +Compute an time $0$ consumption $c_0 $ : $$ c_t = c_0 = \left( \frac{1 - R^{-1}}{1 - R^{-(T+1)}} \right) (a_0 + \sum_{t=0}^T R^t y_t ) , \quad t = 0, 1, \ldots, T @@ -218,9 +233,10 @@ $$ ### Step 3 -In this step, we use the system of equations {eq}`eq:a_t` for $t=0, \ldots, T$ to compute a path $a$ of financial wealth. +Use the system of equations {eq}`eq:a_t` for $t=0, \ldots, T$ to compute a path $a$ of financial wealth. + +To do this, we translate that system of difference equations into a single matrix equation as follows: -To do this, we translated that system of difference equations into a single matrix equation as follows (we'll say more about the mechanics of using linear algebra to solve such difference equations later in the last part of this lecture): $$ \begin{bmatrix} @@ -244,18 +260,18 @@ $$ \begin{bmatrix} a_1 \cr a_2 \cr a_3 \cr \vdots \cr a_T \cr a_{T+1} \end{bmatrix} $$ -It should turn out automatically that + +Because we have built into our calculations that the consumer leaves the model with exactly zero assets, just barely satisfying the +terminal condition that $a_{T+1} \geq 0$, it should turn out that $$ a_{T+1} = 0. $$ + -We have built into the our calculations that the consumer leaves life with exactly zero assets, just barely satisfying the -terminal condition that $a_{T+1} \geq 0$. - -Let's verify this with our Python code. +Let's verify this with Python code. -First we implement this model in `compute_optimal` +First we implement the model with `compute_optimal` ```{code-cell} ipython3 def compute_optimal(model, a0, y_seq): @@ -279,10 +295,14 @@ def compute_optimal(model, a0, y_seq): return c_seq, a_seq ``` -We use an example where the consumer inherits $a_0<0$ (which can be interpreted as a student debt). +We use an example where the consumer inherits $a_0<0$. + +This can be interpreted as a student debt. The non-financial process $\{y_t\}_{t=0}^{T}$ is constant and positive up to $t=45$ and then becomes zero afterward. +The drop in non-financial income late in life reflects retirement from work. + ```{code-cell} ipython3 # Financial wealth a0 = -2 # such as "student debt" @@ -297,7 +317,7 @@ print('check a_T+1=0:', np.abs(a_seq[-1] - 0) <= 1e-8) ``` -The visualization shows the path of non-financial income, consumption, and financial assets. +The graphs below show paths of non-financial income, consumption, and financial assets. ```{code-cell} ipython3 # Sequence Length @@ -314,9 +334,9 @@ plt.ylabel(r'$c_t,y_t,a_t$') plt.show() ``` -Note that $a_{T+1} = 0$ is satisfied. +Note that $a_{T+1} = 0$, as anticipated. -We can further evaluate the welfare using the formula {eq}`welfare` +We can evaluate welfare criterion {eq}`welfare` ```{code-cell} ipython3 def welfare(model, c_seq): @@ -332,12 +352,12 @@ print('Welfare:', welfare(cs_model, c_seq)) ### Feasible consumption variations -Earlier, we had promised to present an argument that supports our claim that a constant consumption play $c_t = c_0$ for all +We promised to justify our claim that a constant consumption play $c_t = c_0$ for all $t$ is optimal. Let's do that now. -Although simple and direct, the approach we'll take is actually an example of what is called the "calculus of variations". +The approach we'll take is an elementary example of the "calculus of variations". Let's dive in and see what the key idea is. @@ -348,18 +368,20 @@ $$ \sum_{t=0}^T R^{-t} v_t = 0 $$ -This equation says that the **present value** of admissible variations must be zero. +This equation says that the **present value** of admissible consumption path variations must be zero. -(So once again, we encounter our formula for the present value of an "asset".) +So once again, we encounter a formula for the present value of an "asset": -Here we'll compute a two-parameter class of admissible variations + * we require that the present value of consumption path variations be zero. + +Here we'll restrict ourselves to a two-parameter class of admissible consumption path variations of the form $$ v_t = \xi_1 \phi^t - \xi_0 $$ -We say two and not three-parameter class because $\xi_0$ will be a function of $(\phi, \xi_1; R)$ that guarantees that the variation is feasible. +We say two and not three-parameter class because $\xi_0$ will be a function of $(\phi, \xi_1; R)$ that guarantees that the variation sequence is feasible. Let's compute that function. @@ -389,13 +411,13 @@ $$ This is our formula for $\xi_0$. -Evidently, if $c^o$ is a budget-feasible consumption path, then so is $c^o + v$, +**Key Idea:** if $c^o$ is a budget-feasible consumption path, then so is $c^o + v$, where $v$ is a budget-feasible variation. Given $R$, we thus have a two parameter class of budget feasible variations $v$ that we can use to compute alternative consumption paths, then evaluate their welfare. -Now let's compute and visualize the variations +Now let's compute and plot consumption path variations variations ```{code-cell} ipython3 def compute_variation(model, ξ1, ϕ, a0, y_seq, verbose=1): @@ -415,7 +437,7 @@ def compute_variation(model, ξ1, ϕ, a0, y_seq, verbose=1): +++ {"user_expressions": []} -We visualize variations with $\xi_1 \in \{.01, .05\}$ and $\phi \in \{.95, 1.02\}$ +We visualize variations for $\xi_1 \in \{.01, .05\}$ and $\phi \in \{.95, 1.02\}$ ```{code-cell} ipython3 fig, ax = plt.subplots() @@ -518,26 +540,25 @@ plt.show() ## Wrapping up the consumption-smoothing model -The consumption-smoothing model of Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`) is a cornerstone of modern macro -that has important ramifications about the size of the Keynesian "fiscal policy multiplier" described briefly in +The consumption-smoothing model of Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`) is a cornerstone of modern macro that has important ramifications for the size of the Keynesian "fiscal policy multiplier" described briefly in quantecon lecture {doc}`geometric series `. -In particular, Milton Friedman and others showed that it **lowered** the fiscal policy multiplier relative to the one implied by -the simple Keynesian consumption function presented in {doc}`geometric series `. +In particular, it **lowers** the government expenditure multiplier relative to one implied by +the original Keynesian consumption function presented in {doc}`geometric series `. -Friedman and Hall's work opened the door to a lively literature on the aggregate consumption function and implied fiscal multipliers that -remains very active today. +Friedman's work opened the door to an enlighening literature on the aggregate consumption function and associated government expenditure multipliers that +remains active today. -## Difference equations with linear algebra +## Appendix: solving difference equations with linear algebra In the preceding sections we have used linear algebra to solve a consumption smoothing model. -The same tools from linear algebra -- matrix multiplication and matrix inversion -- can be used to study many other dynamic models too. +The same tools from linear algebra -- matrix multiplication and matrix inversion -- can be used to study many other dynamic models. -We'll concluse this lecture by giving a couple of examples. +We'll conclude this lecture by giving a couple of examples. -In particular, we'll describe a useful way of representing and "solving" linear difference equations. +We'll describe a useful way of representing and "solving" linear difference equations. To generate some $y$ vectors, we'll just write down a linear difference equation with appropriate initial conditions and then use linear algebra to solve it. @@ -570,10 +591,10 @@ y_1 \cr y_2 \cr y_3 \cr \vdots \cr y_T \begin{bmatrix} \lambda y_0 \cr 0 \cr 0 \cr \vdots \cr 0 \end{bmatrix} -$$ +$$ (eq:first_order_lin_diff) -Multiplying both sides by inverse of the matrix on the left provides the solution +Multiplying both sides of {eq}`eq:first_order_lin_diff` by the inverse of the matrix on the left provides the solution ```{math} :label: fst_ord_inverse @@ -597,7 +618,7 @@ y_1 \cr y_2 \cr y_3 \cr \vdots \cr y_T ```{exercise} :label: consmooth_ex1 -In the {eq}`fst_ord_inverse`, we multiply the inverse of the matrix $A$. In this exercise, please confirm that +To get {eq}`fst_ord_inverse`, we multiplied both sides of {eq}`eq:first_order_lin_diff` by the inverse of the matrix $A$. Please confirm that $$ \begin{bmatrix} diff --git a/lectures/pv.md b/lectures/pv.md index f4678a2a..c9923531 100644 --- a/lectures/pv.md +++ b/lectures/pv.md @@ -18,40 +18,73 @@ kernelspec: This lecture describes the **present value model** that is a starting point of much asset pricing theory. -We'll use the calculations described here in several subsequent lectures. +Asset pricing theory is a component of theories about many economic decisions including -Our only tool is some elementary linear algebra operations, namely, matrix multiplication and matrix inversion. + * consumption + * labor supply + * education choice + * demand for money + +In asset pricing theory, and in economic dynamics more generally, a basic topic is the relationship +among different **time series**. + +A **time series** is a **sequence** indexed by time. + +In this lecture, we'll represent a sequence as a vector. + +So our analysis will typically boil down to studying relationships among vectors. + +Our main tools in this lecture will be + + * matrix multiplication, and + * matrix inversion. + +We'll use the calculations described here in subsequent lectures, including {doc}`consumption smoothing `, {doc}`equalizing difference model `, and +{doc}`monetarist theory of price levels `. Let's dive in. +## Analysis + + + Let * $\{d_t\}_{t=0}^T $ be a sequence of dividends or "payouts" * $\{p_t\}_{t=0}^T $ be a sequence of prices of a claim on the continuation of - the asset stream from date $t$ on, namely, $\{d_s\}_{s=t}^T $ + the asset's payout stream from date $t$ on, namely, $\{d_s\}_{s=t}^T $ * $ \delta \in (0,1) $ be a one-period "discount factor" * $p_{T+1}^*$ be a terminal price of the asset at time $T+1$ We assume that the dividend stream $\{d_t\}_{t=0}^T $ and the terminal price $p_{T+1}^*$ are both exogenous. +This means that they are determined outside the model. + Assume the sequence of asset pricing equations $$ p_t = d_t + \delta p_{t+1}, \quad t = 0, 1, \ldots , T $$ (eq:Euler1) -This is a "cost equals benefits" formula. +We say equation**s**, plural, because there are $T+1$ equations, one for each $t =0, 1, \ldots, T$. + + +Equations {eq}`eq:Euler1` assert that price paid to purchase the asset at time $t$ equals the payout $d_t$ plus the price at time $t+1$ multiplied by a time discount factor $\delta$. -It says that the cost of buying the asset today equals the reward for holding it -for one period (which is the dividend $d_t$) and then selling it, at $t+1$. +Discounting tomorrow's price by multiplying it by $\delta$ accounts for the ``value of waiting one period''. -The future value $p_{t+1}$ is discounted using $\delta$ to shift it to a present value, so it is comparable with $d_t$ and $p_t$. +We want to solve the system of $T+1$ equations {eq}`eq:Euler1` for the asset price sequence $\{p_t\}_{t=0}^T $ as a function of the divident sequence $\{d_t\}_{t=0}^T $ and the exogenous terminal +price $p_{T+1}^*$. -We want to solve for the asset price sequence $\{p_t\}_{t=0}^T $ as a function -of $\{d_t\}_{t=0}^T $ and $p_{T+1}^*$. +A system of equations like {eq}`eq:Euler1` is an example of a linear **difference equation**. -In this lecture we show how this can be done using matrix algebra. +There are powerful mathematical methods available for solving such systems and they are well worth +studying in their own right, being the foundation for the analysis of many interesting economic models. + +For an example, see {doc}`Samuelson multiplier-accelerator ` + +In this lecture, we'll solve system {eq}`eq:Euler1` using matrix multiplication and matrix inversion, basic tools from linear algebra introduced in {doc}`linear equations and matrix algebra `. We will use the following imports @@ -64,9 +97,9 @@ import matplotlib.pyplot as plt +++ -## Present value calculations +## Representing sequences as vectors -The equations in [](eq:Euler1) can be stacked, as in +The equations in system {eq}`eq:Euler1` can be arranged as follows: $$ \begin{aligned} @@ -109,7 +142,7 @@ recover the equations in [](eq:Euler_stack). ```{exercise-end} ``` -In vector-matrix notation, we can write the system [](eq:pvpieq) as +In vector-matrix notation, we can write system {eq}`eq:pvpieq` as $$ A p = d + b @@ -143,19 +176,20 @@ $$ \end{bmatrix} $$ -The solution for prices is given by +The solution for the vector of prices is $$ p = A^{-1}(d + b) $$ (eq:apdb_sol) -Here is a small example, where the dividend stream is given by +For example, suppose that the dividend stream is $$ d_{t+1} = 1.05 d_t, \quad t = 0, 1, \ldots , T-1. $$ +Let's write Python code to compute and plot the dividend stream. ```{code-cell} ipython3 T = 6 @@ -171,6 +205,7 @@ ax.legend() ax.set_xlabel('time') plt.show() ``` +Now let's compute and plot the asset price. We set $\delta$ and $p_{T+1}^*$ to @@ -198,7 +233,7 @@ Let's inspect $A$ A ``` -Now let's solve for prices using [](eq:apdb_sol). +Now let's solve for prices using {eq}`eq:apdb_sol`. ```{code-cell} ipython3 b = np.zeros(T+1) @@ -212,7 +247,7 @@ plt.show() ``` -We can also consider a cyclically growing dividend sequence, such as +Now let's consider a cyclically growing dividend sequence: $$ d_{t+1} = 1.01 d_t + 0.1 \sin t, \quad t = 0, 1, \ldots , T-1. @@ -248,7 +283,7 @@ Compute the corresponding asset price sequence when $p^*_{T+1} = 0$ and $\delta :class: dropdown ``` -We proceed as above after modifying parameters and $A$. +We proceed as above after modifying parameters and consequently the matrix $A$. ```{code-cell} ipython3 δ = 0.98 @@ -281,6 +316,8 @@ eliminates the cycles. ## Analytical expressions +By the [inverse matrix theorem](https://en.wikipedia.org/wiki/Invertible_matrix), a matrix $B$ is the inverse of $A$ whenever $A B$ is the identity. + It can be verified that the inverse of the matrix $A$ in {eq}`eq:pvpieq` is @@ -302,13 +339,13 @@ $$ (eq:Ainv) Check this by showing that $A A^{-1}$ is equal to the identity matrix. -(By the [inverse matrix theorem](https://en.wikipedia.org/wiki/Invertible_matrix), a matrix $B$ is the inverse of $A$ whenever $A B$ is the identity.) + ```{exercise-end} ``` -If we use the expression [](eq:Ainv) in [](eq:apdb_sol) and perform the indicated matrix multiplication, we shall find that +If we use the expression {eq}`eq:Ainv` in {eq}`eq:apdb_sol` and perform the indicated matrix multiplication, we shall find that $$ p_t = \sum_{s=t}^T \delta^{s-t} d_s + \delta^{T+1-t} p_{T+1}^* @@ -317,12 +354,12 @@ $$ (eq:ptpveq) Pricing formula {eq}`eq:ptpveq` asserts that two components sum to the asset price $p_t$: - * a **fundamental component** $\sum_{s=t}^T \delta^{s-t} d_s$ that equals the discounted present value of prospective dividends + * a **fundamental component** $\sum_{s=t}^T \delta^{s-t} d_s$ that equals the **discounted present value** of prospective dividends * a **bubble component** $\delta^{T+1-t} p_{T+1}^*$ The fundamental component is pinned down by the discount factor $\delta$ and the -"fundamentals" of the asset (in this case, the dividends). +payout of the asset (in this case, dividends). The bubble component is the part of the price that is not pinned down by fundamentals. @@ -343,7 +380,7 @@ $$ ## More about bubbles -For a few moments, let's focus on the special case of an asset that will never pay dividends, in which case +For a few moments, let's focus on the special case of an asset that never pays dividends, in which case $$ \begin{bmatrix} @@ -385,8 +422,9 @@ $$ (eq:eqbubbleterm) for some positive constant $c$. -In this case, it can be verified that when we multiply both sides of {eq}`eq:pieq2` by -the matrix $A^{-1}$ presented in equation {eq}`eq:Ainv`, we shall find that +In this case, when we multiply both sides of {eq}`eq:pieq2` by +the matrix $A^{-1}$ presented in equation {eq}`eq:Ainv`, we + find that $$ p_t = c \delta^{-t} @@ -402,11 +440,10 @@ $$ R_t = \frac{p_{t+1}}{p_t} $$ (eq:rateofreturn) -Equation {eq}`eq:bubble` confirms that an asset whose sole source of value is a bubble -that earns a gross rate of return +Substituting equation {eq}`eq:bubble` into equation {eq}`eq:rateofreturn` confirms that an asset whose sole source of value is a bubble earns a gross rate of return $$ -R_t = \delta^{-1} > 1 . +R_t = \delta^{-1} > 1 , t = 0, 1, \ldots, T $$ @@ -417,7 +454,7 @@ $$ :label: pv_ex_a ``` -Give analytical expressions for the asset price $p_t$ under the +Give analytical expressions for an asset price $p_t$ under the following settings for $d$ and $p_{T+1}^*$: 1. $p_{T+1}^* = 0, d_t = g^t d_0$ (a modified version of the Gordon growth formula)