# Workout of Dynamic Programming using Bellman Equation

I am studying "dynamic programming" in macroeconomics with respect to the Neo-Classical Growth Model. The following is the Dynamic Problem using Bellman Equation I have worked out. Would love your input on the steps and if there is any advice that could simplify calculations or a way to check the solution another way.

Consider the following planning problem:

subject to

where " $zk_t$ (k - is the capital input, and z is the productivity parameter) is the production function of a representative firm of the form $f(k)$ where we assume perfectly inelastic labour supply, i.e., households don't value leisure and that means labour supply is equal to one. Symbolically, $F(K,N) = F(K,1) = f(k)$

$\textbf{My Workout}$

(a)The dynamic programming problem can be written via the Bellman Equation :

$V(k) = max_{0\geq k'\leq zk}\{ \frac{(zk+(1-\delta)k -k')^{(1-\sigma)}}{1-\sigma} + \beta V(k') \}$

where $V(k')$ is the value function for future value of capital $(k')$ .

(b) Using the Guess and Verify method we can solve for the value of $(k')$ We make a guess of :

$V(k) = \frac{Ak^{1-\sigma}}{1-\sigma}$ where $A$ is the undetermined coefficient.

Step 1: Solving the maximisation problem above assuming our guess is correct:

$max_{0\geq k'\leq zk}\{ \frac{(zk+(1-\delta)k -k')^{(1-\sigma)}}{1-\sigma} + \beta \frac{Ak'^{(1-\sigma)}}{1-\sigma} \}$

The first order condition with respect to $k'$:, gives the following equation for $k'$ :

$k' = (z+1-\delta)k\frac{(\beta A)^{1/\sigma}}{1+(\beta A)^{1/\sigma}}$

Step 2: We use this value of $k'$ and plug it back into the Bellman Equation and solve for the undetermined coefficient $A$:

$V(k) = \{ \frac{(zk+(1-\delta)k -k')^{(1-\sigma)}}{1-\sigma} + \beta \frac{Ak'^{(1-\sigma)}}{1-\sigma} \}$

After substituting $k'$ above , this becomes:

$\frac{Ak^{(1-\sigma)}}{1-\sigma} = \frac{k^{(1-\sigma)}}{1-\sigma} [ (z+1-\delta)^{(1-\sigma)}(1+(\beta A)^{1/\sigma})^{\sigma} ]$

Here we see the term in the $[.]$ is independent of $k$. The term in $[.]$ is of the form of coefficient A on the left hand side of the equation. This implies , that :

$A = (z+1-\delta)^{(1-\sigma)}(1+(\beta A)^{1/\sigma})^{\sigma}$

Solving for $A$, gives us:

$A = \frac{(z+1-\delta)^{(1-\sigma)}}{(1-(z+1-\delta)^{\frac{(1-\sigma)}{\sigma}}\beta^{\frac{1}{\sigma}})^{\sigma}}$

This verifies our initial guess is correct , since $A$ is independent of $k$. Now we replace $A$ in the equation of $k'$ and get:

$k' = \beta^{\frac{1}{\sigma}}(z+1-\delta)^{\frac{1}{\sigma}}k$

This equation of $k'$ shows us how the economy evolves. And the policy function for $k$, " $g(k)$ " is given by this equation above.

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The above is my workout of the dynamic programming problem. I would love your comment and advice if there is some way I can improve my understanding of solving dynamic programming problem.

Looking forward to the discussion.

• what is $zk_t$supposed to stand for? – user14471 Oct 9 '17 at 19:40
• @user43282 $zk_t$ is the production function of the form F(K,N) where we assume perfectly inelastic labour supply as households don't value leisure so $(N=1)$ . Hence, $zk_t$ is the capital input times the parameter (productivity) "z". Hope this clears up. I'll edit it and mention it. Thanks. – Sky Oct 10 '17 at 12:13
• when the general form of the production function is $Y=F(K,N)$ and we assume constant returns to scale, then $y=F(k,1)\equiv f(k)$ is output per labor input (average productivity) where $y\equiv Y/N$ and $k\equiv K/N$; when $f(k_t)=z k_t$ this is effectively the same as using a linear technology; are you purposefully using a linear technology? – user14471 Oct 10 '17 at 13:02
• @user43282 thanks for explaining it a little more. Yes, I am purposefully assuming a linear technology for simplification of a planning problem. Thanks. – Sky Oct 10 '17 at 16:44