You should inclConsider the problem of a value maximizing firm whose profit function at time t is given by

$\backslash$Pi $($Kt$)=$ e

$$AKt

$\backslash$theta

where e is the natural exponent, $$A denotes fixed the productivity level, Kt denotes capital and

$\backslash$theta is a parameter representing the elasticity of output with respect to capital. Assume that there

is no depreciation of capital, so the law of motion of capital is given as

Kt$+1=$ Kt $+$ It

where It

is investments in capital at time t$.$ The price of a unit of capital good is p and investment

is subject to a smooth convex installation cost given by

C$($It

$,$ Kt$)=\backslash$gamma

$2$

I

$2$

t

$,$

where $\backslash$gamma is a constant denoting the adjustment cost parameter. Time is discrete and runs to

infinity, t $=0,1...\backslash$infty $.$ Firm manager discounts future values with the factor $\backslash$beta $.$

Based on the above information, answer the following two questions:

$1\mathrm{.}1$ Write down the Bellman equation and derive the optimal investment decision condition.

Define marginal Q and provide an economic interpretation. How is average Q related to

marginal Q for this firm? $[20\%]$

$1\mathrm{.}2$ Solve the Bellman equation using dynamic programming. You can calibrate model parameters as follows

$\backslash$beta $=0\mathrm{.}95,\backslash$theta $=0\mathrm{.}7,\backslash$gamma $=0\mathrm{.}2$ and p $=1\mathrm{.}2.$ Discretize capital grid with $301$ uniformly

spaced points in the interval $[30,80].$ Constant Productivity Level is $$A $=1\mathrm{.}5.$

Plot value function and investment policy functions. Interpret these graphs. Explain how

optimal investment responds to

$$ changes in the adjustment cost parameter.

$$ changes in how the firm manager values time.Consider the problem of a value maximizing firm whose profit function at time $t$ is given by

$(K_t)=e^{\frac{?}{\text{b}\text{a}\text{r}}(A)}{K}_t^{}$

where $e$ is the natural exponent, $\frac{?}{\text{b}\text{a}\text{r}}(A)$ denotes fixed the productivity level, $K_t$ denotes capital and

$$ is a parameter representing the elasticity of output with respect to capital. Assume that there

is no depreciation of capital, so the law of motion of capital is given as

$K_{t+1}=K_t+I_t$

where $I_t$ is investments in capital at time $t.$ The price of a unit of capital good is $p$ and investment

is subject to a smooth convex installation cost given by

$C(I_t,K_t)=\frac{}{2}{I}_t^2,$

where $$ is a constant denoting the adjustment cost parameter. Time is discrete and runs to

infinity, $t=0,1$dots$.$ Firm manager discounts future values with the factor $.$

Based on the above information, answer the following two questions:

$1\mathrm{.}1$ Write down the Bellman equation and derive the optimal investment decision condition.

Define marginal $Q$ and provide an economic interpretation. How is average Q related to

marginal Q for this firm?

$20\%$

$1\mathrm{.}2$ Solve the Bellman equation using dynamic programming. You can calibrate model param$-$

eters as follows

$=0\mathrm{.}95,=0\mathrm{.}7,=0\mathrm{.}2$ and $p=1\mathrm{.}2.$ Discretize capital grid with $301$ uniformly

spaced points in the interval $30,80.$ Constant Productivity Level is $\frac{?}{\text{b}\text{a}\text{r}}(A)=1\mathrm{.}5.$

Plot value function and investment policy functions. Interpret these graphs. Explain how

optimal investment responds to

changes in the adjustment cost parameter.

changes in how the firm manager values time.Consider the problem of a value maximizing firm whose profit function at time $t$ is given by

$(K_t)=e^{\frac{?}{\text{b}\text{a}\text{r}}(A)}{K}_t^{}$

where $e$ is the natural exponent, $\frac{?}{\text{b}\text{a}\text{r}}(A)$ denotes fixed the productivity level, $K_t$ denotes capital and

$$ is a parameter representing the elasticity of output with respect to capital. Assume that there

is no depreciation of capital, so the law of motion of capital is given as

$K_{t+1}=K_t+I_t$

where $I_t$ is investments in capital at time $t.$ The price of a unit of capital good is $p$ and investment

is subject to a smooth convex installation cost given by

$C(I_t,K_t)=\frac{}{2}{I}_t^2$

where $$ is a constant denoting the adjustment cost parameter. Time is discrete and runs to

infinity, $t=0,1$dots$.$ Firm manager discounts future values with the factor