<p>The way you would go about solving this problem is as the ChinG said is by setting up the Hamiltonian. In this case this is:</p>
<p><span class="math-container">$$\mathcal{H}:e^{-\rho t} \frac{C(t)^{1-\theta}-1}{1-\theta}+\mu(t)\left[Y(t)-C(t)-X(t)-Z(t)\right]$$</span></p>
<p>Taking the first order condition for this problem we get:
<span class="math-container">$$\frac{\partial \mathcal{H}}{\partial C(t)}:e^{-\rho t}C(t)^{-\theta}-\mu(t)=0 $$</span>
or
<span class="math-container">$$e^{-\rho t}C(t)^{-\theta}=\mu(t)$$</span></p>
<p>Taking the natural logs of this equation and differentiating with respect to time we get:
<span class="math-container">$$\ln\left[\frac{\partial \mathcal{H}}{\partial C(t)}\right]:-\rho t -\theta \ln [C(t)]=\ln[\mu(t)]$$</span>
<span class="math-container">$$\frac{\partial\ln\left[\mathcal{H}_{C(t)}\right]}{\partial t}:-\rho-\theta\frac{\dot{C}(t)}{C(t)}=-\frac{\dot{\mu}(t)}{\mu(t)}$$</span>
Rearranging the above and noting that <span class="math-container">$\frac{\dot{\mu}(t)}{\mu(t)}=r(t)$</span>
we get:
<span class="math-container">$$\frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r(t)-\rho)$$</span>
Which is the Euler equation.</p>
<p>Hope this helps.</p>

CC BY-SA 4.0
2020-07-08T14:55:18.543Z
Hi, thanks for the response! On thing I don't quite understand is why $\dot{\mu}/\mu = r(t)$?
Finalblue
CC BY-SA 4.0
2020-07-09T01:37:08.593Z
@Finalblue Theres a ton of math to get that result so I tried to be sneaky and handwave a bit. I could tell you that intuitively we think of $\mu$ as the marginal utility from output over time. We interpret the growth in that utility from output over time $\dot{\mu}/\mu$ as the rate of return on the market $r(t)$.
EconJohn
CC BY-SA 4.0
2020-07-08T01:41:41.100Z
2020-07-09T01:28:03.047Z
2020-07-09T01:28:03.047Z
EconJohn
<p>In the standard expanding variety model where the representative household faces the following CRRA utility function</p>
<p><span class="math-container">$$
\int_{0}^{\infty} \exp(-\rho t) \frac{C(t)^{1-\theta}-1}{1-\theta}dt
$$</span></p>
<p>and final goods that is produced competitively with the following production
<span class="math-container">$$
Y(t) = \frac{1}{1-\beta} \left( \int_{0}^{N(t)}x(v,t)^{1-\beta}dv\right)L^{\beta}
$$</span></p>
<p>with the following resource constraint</p>
<p><span class="math-container">$$
C(t) + X(t) + Z(t) \leq Y(t)
$$</span></p>
<p>where <span class="math-container">$X(t)$</span> is the total spending on machines, and <span class="math-container">$Z(t)$</span> is the total expenditure on research and development.</p>
<p>The innovation possibilities frontier takes the form</p>
<p><span class="math-container">$$
\dot{N}(t) = \eta Z(t)
$$</span></p>
<p>The intermediary goods firm faces the following problem where</p>
<p><span class="math-container">$$
r(t) V(v,t) - \dot{V}(v,t) = \pi(v,t)
$$</span></p>
<p>The thing I don't understand about the model is how the Euler Equation is derived. In almost everywhere I look the Euler Equation is the standard</p>
<p><span class="math-container">$$
\frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta}(r(t)-\rho)
$$</span></p>
<p>I don't understand how you would derive this equation.</p>

CC BY-SA 4.0
2020-07-05T17:19:52.813Z
Hint: set up the Hamiltonian
ChinG
<macroeconomics><endogenous-growth>
How to derive the standard euler equation from the Expanding Variety Model
Finalblue
# How to derive the standard euler equation from the Expanding Variety Model

(Asked by: username_0 on 2020-07-04)

In the standard expanding variety model where the representative household faces the following CRRA utility function

$$
\int\_{0}^{\infty} \exp(-\rho t) \frac{C(t)^{1-\theta}-1}{1-\theta}dt
$$

and final goods that is produced competitively with the following production
$$
Y(t) = \frac{1}{1-\beta} \left( \int\_{0}^{N(t)}x(v,t)^{1-\beta}dv\right)L^{\beta}
$$

with the following resource constraint

$$
C(t) + X(t) + Z(t) \leq Y(t)
$$

where $X(t)$ is the total spending on machines, and $Z(t)$ is the total expenditure on research and development.

The innovation possibilities frontier takes the form

$$
\dot{N}(t) = \eta Z(t)
$$

The intermediary goods firm faces the following problem where

$$
r(t) V(v,t) - \dot{V}(v,t) = \pi(v,t)
$$

The thing I don't understand about the model is how the Euler Equation is derived. In almost everywhere I look the Euler Equation is the standard

$$
\frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta}(r(t)-\rho)
$$

I don't understand how you would derive this equation.

**Tags:** macroeconomics, endogenous-growth
> **Comments:**
> **username_1**: Hint: set up the Hamiltonian  

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## 1 answers

### Answer by username_2 on 2020-07-08. Score: 2

The way you would go about solving this problem is as the ChinG said is by setting up the Hamiltonian. In this case this is:

$$\mathcal{H}:e^{-\rho t} \frac{C(t)^{1-\theta}-1}{1-\theta}+\mu(t)\left[Y(t)-C(t)-X(t)-Z(t)\right]$$

Taking the first order condition for this problem we get:
$$\frac{\partial \mathcal{H}}{\partial C(t)}:e^{-\rho t}C(t)^{-\theta}-\mu(t)=0 $$
or
$$e^{-\rho t}C(t)^{-\theta}=\mu(t)$$

Taking the natural logs of this equation and differentiating with respect to time we get:
$$\ln\left[\frac{\partial \mathcal{H}}{\partial C(t)}\right]:-\rho t -\theta \ln [C(t)]=\ln[\mu(t)]$$
$$\frac{\partial\ln\left[\mathcal{H}\_{C(t)}\right]}{\partial t}:-\rho-\theta\frac{\dot{C}(t)}{C(t)}=-\frac{\dot{\mu}(t)}{\mu(t)}$$
Rearranging the above and noting that $\frac{\dot{\mu}(t)}{\mu(t)}=r(t)$
we get:
$$\frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r(t)-\rho)$$
Which is the Euler equation.

Hope this helps.

> **Comments:**
> **username_0**: Hi, thanks for the response! On thing I don't quite understand is why $\dot{\mu}/\mu = r(t)$?  
> **username_2**: @Finalblue Theres a ton of math to get that result so I tried to be sneaky and handwave a bit. I could tell you that intuitively we think of $\mu$ as the marginal utility from output over time. We interpret the growth in that utility from output over time $\dot{\mu}/\mu$ as the rate of return on the market $r(t)$.  

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