@@ -66,27 +66,27 @@ from cmath import sqrt
6666
6767### Samuelson's model
6868
69- Samuelson used a * second-order linear difference equation* to
69+ Samuelson used a ** second-order linear difference equation* * to
7070represent a model of national output based on three components:
7171
72- - a * national output identity* asserting that national output or national income is the
72+ - a ** national output identity* * asserting that national output or national income is the
7373 sum of consumption plus investment plus government purchases.
74- - a Keynesian * consumption function* asserting that consumption at
74+ - a Keynesian ** consumption function* * asserting that consumption at
7575 time $t$ is equal to a constant times national output at time $t-1$.
76- - an investment * accelerator* asserting that investment at time
77- $t$ equals a constant called the * accelerator coefficient*
76+ - an investment ** accelerator* * asserting that investment at time
77+ $t$ equals a constant called the ** accelerator coefficient* *
7878 times the difference in output between period $t-1$ and
7979 $t-2$.
8080
8181Consumption plus investment plus government purchases
82- constitute * aggregate demand,* which automatically calls forth an
83- equal amount of * aggregate supply* .
82+ constitute ** aggregate demand,* * which automatically calls forth an
83+ equal amount of ** aggregate supply* * .
8484
8585(To read about linear difference equations see [ here] ( https://en.wikipedia.org/wiki/Linear_difference_equation ) or chapter IX of {cite}` Sargent1987 ` .)
8686
8787Samuelson used the model to analyze how particular values of the
8888marginal propensity to consume and the accelerator coefficient might
89- give rise to transient * business cycles* in national output.
89+ give rise to transient ** business cycles* * in national output.
9090
9191Possible dynamic properties include
9292
@@ -100,7 +100,7 @@ adds a random shock to the right side of the national income
100100identity representing random fluctuations in aggregate demand.
101101
102102This modification makes national output become governed by a second-order
103- * stochastic linear difference equation* that, with appropriate parameter values,
103+ ** stochastic linear difference equation* * that, with appropriate parameter values,
104104gives rise to recurrent irregular business cycles.
105105
106106(To read about stochastic linear difference equations see chapter XI of
@@ -152,7 +152,7 @@ and the national income identity
152152Y_t = C_t + I_t + G_t
153153```
154154
155- - The parameter $\alpha$ is peoples' * marginal propensity to consume*
155+ - The parameter $\alpha$ is peoples' ** marginal propensity to consume* *
156156 out of income - equation {eq}` consumption ` asserts that people consume a fraction of
157157 $\alpha \in (0,1)$ of each additional dollar of income.
158158- The parameter $\beta > 0$ is the investment accelerator coefficient - equation
@@ -193,7 +193,7 @@ a constant value as $t$ becomes large.
193193We are interested in studying
194194
195195- the transient fluctuations in $Y_t$ as it converges to its
196- * steady state* level
196+ ** steady state* * level
197197- the * rate* at which it converges to a steady state level
198198
199199The deterministic version of the model described so far --- meaning that
@@ -204,10 +204,10 @@ fluctuations by adding a random shock to aggregate demand.
204204
205205### Stochastic version of the model
206206
207- We create a * random* or * stochastic* version of the model by adding
208- a random process of * shocks* or * disturbances*
207+ We create a ** random** or ** stochastic* * version of the model by adding
208+ a random process of ** shocks** or ** disturbances* *
209209$\{ \sigma \epsilon_t \} $ to the right side of equation {eq}` second_order ` ,
210- leading to the * second-order scalar linear stochastic difference equation* :
210+ leading to the ** second-order scalar linear stochastic difference equation* * :
211211
212212``` {math}
213213:label: second_stochastic
@@ -235,7 +235,7 @@ Y_{t+2} - \rho_1 Y_{t+1} - \rho_2 Y_t = 0
235235```
236236
237237To discover the properties of the solution of {eq}` second_stochastic2 ` ,
238- it is useful first to form the * characteristic polynomial*
238+ it is useful first to form the ** characteristic polynomial* *
239239for {eq}` second_stochastic2 ` :
240240
241241``` {math}
@@ -246,7 +246,7 @@ z^2 - \rho_1 z - \rho_2
246246
247247where $z$ is possibly a complex number.
248248
249- We want to find the two * zeros* (a.k.a. * roots* ) -- namely
249+ We want to find the two ** zeros** (a.k.a. ** roots* * ) -- namely
250250$\lambda_1, \lambda_2$ -- of the characteristic polynomial.
251251
252252These are two special values of $z$, say $z= \lambda_1$ and
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