In this problem, we study a growth model in which technological progress is endogenous. Production and capital accumulation take place as in a standard Solow growth model:

$Y_t=K_t^{}(A_t{L}_{Y_t}{)}^{1-}$

$K_{t+1}=(1-)K_t+Y_t$

A fraction ${}_A$ of the population is involved in the production of new ideas and a fraction $1-{}_A$ is involved in the production of output. Hence,

$L_{Y_t}=(1-{}_A)L_t$

$L_{At}={}_A{L}_t,$

where $L_t$ is the population at date $t.$ The population grows at the rate $,$ so $L_{t+1}=(1+)L_t.$ Finally, we assume that new ideas are produced as follows:

$A_{t+1}-A_t=\frac{1}{}{A}_t^{}{L}_{At}^{},$ where and $<mn\; id="EditorElement\_1179072">1</mn>$

a$.$ Derive an expression for the steady state growth rate of $A_t,$ call it $\frac{?}{\text{b}\text{a}\text{r}}(g).$

b$.$ As in Assignment $3,$ let $=0\mathrm{.}2,=0\mathrm{.}33$ and $=\mathrm{.}08.$ Suppose population grows at a rate of $1\mathrm{.}5$ percent per year and that per capita output grows at a rate of $1\mathrm{.}5$ percent per year along a steady state growth path $($also as in Assignment $3).$ Also, let ${}_A=\mathrm{.}05.$ What value for $\frac{?}{\text{b}\text{a}\text{r}}(g)$ is consistent with these facts? In addition, assume that $=0$ and $=1.$ Did the growth rate of per capita income play any role in determining the value for $\frac{?}{\text{b}\text{a}\text{r}}(g)?$ In any event, does the calibrated model imply per capita output growth of $1\mathrm{.}5$ percent per year as in the data? Explain.

c$.$ Let $=10,A_0=100$ and $L_0=1000.$ Simulate the path of $g_t$ for $250$ periods. Print a plot of your simulation.

d$.$ Assuming the same parameter values and same $L_0,$ find the value of $A_0$ so that the economy is in steady state in period $0.$ Using this value of $A_0$ instead of the one used in part $($c$)-$that is$,$ assume that we start in steady state$-$suppose that ${}_A$ was increased to $0\mathrm{.}1$ in period $5.$ Create a plot showing what happens to $g_t$ in response to this change.

e$.$ Now, expand the simulation in part $($d$)$ to include other elements of the Solow growth model. Assume, as in part $($d$),$ the economy is initially in a steady state. This requires that you find the steady state for the other elements of the model, in particular the stock of capital $($or$,$ more precisely, $\frac{K_t}{A_t{L}_t}).$ Provide a plot that shows what happens to the log of per capita output for $50$ periods following the change in ${}_A.$ In the same plot, show the log of $y$ with no change in ${}_A.$