Guess and Verify
A Simple Representative Growth Model
Brock and Mirman (1972)
- The social planner solves the following problem: \[ \begin{gathered} \max _{\left\{c_{t}, k_{t+1}\right\}} \sum_{t=0}^{\infty} \beta^{t} \log \left(c_{t}\right) \\ c_{t}+k_{t+1} \leq A k_{t}^{\alpha} \\ k_{0} \text{ given, } c_{t}, k_{t} \geq 0 \text{ for all } t \end{gathered} \]
- The controls are \(c_{t}\) and \(k_{t+1}\), and the only state variable is \(k_{t}\). We can reformulate this problem as a dynamic programming problem as follows: \[ V(k) = \max_{k'} \left\{ \log \left( A k^{\alpha} - k' \right) + \beta V(k') \right\} \tag{1}\]
- If we solve for the capital stock policy rule \(k' = g_k(k)\), we can obtain the policy function for consumption \(c = g_c(k)\) directly from the resource constraint: \[ g_c(k) = A k^{\alpha} - g_k(k) \]
Step 1: Guess a solution
Given \[ u(c) = \log(c) \]
Guess a solution for the value function: \[ V(k) = a + b\log(k) \] where \(a,b\in\mathbb{R}\) are constants.
Step 2: Verify the solution
Substitute the guess into the Bellman equation Equation 1:
\[ a + b \log (k) = \max_{k'} \left\{ \log \left( A k^{\alpha} - k' \right) + \beta (a + b \log (k')) \right\} \]
Consider the first-order condition: \[ \frac{∂}{∂k'} \left\{ \log \left( A k^{\alpha} - k' \right) + \beta (a + b \log (k')) \right\} = 0 \]
That is, \[ -\frac{1}{A k^{\alpha} - k'} + \beta b \frac{1}{k'} = 0 \]
We can then solve for \(k'\): \[ k' = \frac{\beta b}{\beta b + 1} A k^{\alpha} \tag{2}\]
By the Envelope Theorem, we have:
\[ V'(k) = \frac{∂}{∂k} \log ( A k^{\alpha} - k' ) = \frac{A \alpha k^{\alpha-1}}{A k^{\alpha} - k'} \]
Given the form \(V(k) = a + b\log(k)\), we have: \[ V'(k) = \frac{b}{k} \]
So, \[ \frac{A \alpha k^{\alpha-1}}{A k^{\alpha} - k'} = \frac{b}{k} \]
We can then solve for \(k'\) again: \[ k' = (1- \frac{\alpha}{b}) A k^{\alpha} \tag{3}\]
Combining Equation 2 and Equation 3, we obtain: \[ \frac{\beta b}{\beta b + 1} = (1- \frac{\alpha}{b}) \]
Solving for \(b\), we get: \[ b = \frac{\alpha}{1-\alpha \beta} \]
Step 3: Solve for the policy function
Given \(b\) and Equation 2, we can solve for the optimal capital stock \(k'\) as follows:
\[ g_k(k) =k' = \alpha \beta A k^{\alpha} \]
and therefore the policy function for consumption is: \[ g_c(k) = A k ^{\alpha} - g_k(k) = (1-\alpha \beta) A k^{\alpha} \]
Step 4: Finalize the solution
We then solve for \(a\) by substituting \(g_k(k)\) and \(g_c(k)\) into the value function: \[ a + b \log(k) = \log((1-\alpha \beta) A k^{\alpha}) + \beta \left( a + b \log(\alpha \beta A k^{\alpha}) \right) \]
For the RHS, we have: \[ \begin{aligned} \log((1-\alpha \beta) A k^{\alpha}) + \beta \left( a + b \log(\alpha \beta A k^{\alpha}) \right) & = \alpha \log (k) + \alpha \beta b \log (k) + \log((1-\alpha \beta) A) + \beta a + \beta b \log (\alpha \beta A) \\ & = [\alpha + \alpha \beta b ] \log (k) + [\beta a] + [\log((1-\alpha \beta) A) + \beta b \log (\alpha \beta A)] \\ & = b \log (k) + [\beta a] + [\log((1-\alpha \beta) A) + \beta b \log (\alpha \beta A)] \end{aligned} \] By comparing the coefficients of \(\log(k)\) and the constant term, we get: \[ a = [\beta a] + [\log((1-\alpha \beta) A) + \beta b \log (\alpha \beta A)] \]
Thus, we can solve for \(a\): \[ a=\frac{\log((1-\alpha\beta)A)+\beta b\log(\alpha\beta A)}{1-\beta} \]