<p>Consider the risk free steady state. In that situation the Euler equation becomes:</p>

<p>$$ \left(\frac{C_{t+1}-b \overline{C_{t}}}{C_t-b \overline{C_{t-1}}}\right)^{-\sigma}=\frac{1}{\beta \cdot (1+r)}$$</p>

<p>Which implies</p>

<p>$$ \Rightarrow  \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] = \frac{\ln[\beta] + \ln[1+r]}{\sigma}$$</p>

<p>Assume that in the non-stochastic setting that growth is constant of the form:
$$ C_{t+1} = \gamma \cdot C_{t}$$ </p>

<p>Note that by definition
$$ \overline{C_{t}} \equiv \frac{1}{t}\sum^t_{\tau=1} C_{\tau}$$
Under a constant growth rate
$$ \Rightarrow \overline{C_{t}}=\frac{1}{t}\sum^t_{\tau=1} \gamma^{\tau-t}C_{t} = \frac{C_t}{t} (1+\frac{1}{\gamma-1}) = \frac{C_t}{t} (\frac{\gamma}{\gamma-1})$$
$$ \Rightarrow \overline{C_{t+1}}= \frac{C_{t+1}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \frac{C_{t}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \overline{C_{t}}$$</p>

<p>$$ \Rightarrow  \frac{\ln[\beta] + \ln[1+r]}{\sigma} = \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] $$
$$ = \ln[\gamma C_{t}-b \gamma \overline{C_{t-1}}] -\ln[C_t-b \overline{C_{t-1}}] = \ln[\gamma]$$</p>

<p>Therefore the one period growth rate in consumption ($\gamma$) in the non-stochastic model is $$ \gamma = e^{\frac{\ln[\beta] + \ln[1+r]}{\sigma}}  = [\beta(1+r)]^\sigma$$</p>

CC BY-SA 3.0
2017-02-21T19:03:46.563Z
2017-02-21T19:03:46.563Z
BKay
<p>In the following open-economy RBC model consumption level ($C_t$) is not the most important for the representative households, but the difference of the consumption level and the earlier level of the average consumption ($\overline{C_{t-1}}$):
$$H_t=C_t-b \overline{C_{t-1}}$$
$$0&lt;b&lt;1$$</p>

<p>The consumer's utility function:
$$E_0 \sum\limits_{t=1}^{\infty}\beta^{t-1}\left( \frac{H_t^{1-\sigma}}{1-\sigma}+\frac{L_t^{1+\eta}}{1+\eta}\right)$$</p>

<p>After solving the Utility Maximization Problem, deriving the First Order Conditions, here's the Euler-equation:
$$\beta E_t \left(\frac{C_{t+1}-b \overline{C_{t}}}{C_t-b \overline{C_{t-1}}}\right)^{-\sigma}(1+r_t)=1$$</p>

<p><strong>Question:</strong> I've got no idea what I should do with the average of the consumption ($\overline{C_{t}}$) term. How can I calculate with this? I need to find the steady state, and do the log-linearization, but I don't understand the substance of this variable.</p>

CC BY-SA 3.0
2016-10-31T16:57:12.173Z
If this is a representative consumer model, what is the meaning of "average" consumption? How does it differ from actual consumption?
Alecos Papadopoulos
CC BY-SA 3.0
2016-10-31T17:02:03.833Z
I have no idea. Do you think it's just a trick with the notation, Alecos?
Übel Yildmar
CC BY-SA 3.0
2016-10-31T17:07:21.433Z
I think it makes no sense to talk about average consumption. The whole formulation is a familiar "habit formation" variant of the usual model, but using past actual consumption, not past average consumption.
Alecos Papadopoulos
<macroeconomics><open-economy-macro><dynare><real-business-cycles>
How can I calculate with the average of the consumption in an RBC framework?
Übel Yildmar
# How can I calculate with the average of the consumption in an RBC framework?

(Asked by: username_0 on 2016-10-31)

In the following open-economy RBC model consumption level ($C\_t$) is not the most important for the representative households, but the difference of the consumption level and the earlier level of the average consumption ($\overline{C\_{t-1}}$):
$$H\_t=C\_t-b \overline{C\_{t-1}}$$
$$0

The consumer's utility function:
$$E\_0 \sum\limits\_{t=1}^{\infty}\beta^{t-1}\left( \frac{H\_t^{1-\sigma}}{1-\sigma}+\frac{L\_t^{1+\eta}}{1+\eta}\right)$$

After solving the Utility Maximization Problem, deriving the First Order Conditions, here's the Euler-equation:
$$\beta E\_t \left(\frac{C\_{t+1}-b \overline{C\_{t}}}{C\_t-b \overline{C\_{t-1}}}\right)^{-\sigma}(1+r\_t)=1$$

**Question:** I've got no idea what I should do with the average of the consumption ($\overline{C\_{t}}$) term. How can I calculate with this? I need to find the steady state, and do the log-linearization, but I don't understand the substance of this variable.

**Tags:** macroeconomics, open-economy-macro, dynare, real-business-cycles
> **Comments:**
> **username_1**: If this is a representative consumer model, what is the meaning of "average" consumption? How does it differ from actual consumption?  
> **username_0**: I have no idea. Do you think it's just a trick with the notation, Alecos?  
> **username_1**: I think it makes no sense to talk about average consumption. The whole formulation is a familiar "habit formation" variant of the usual model, but using past actual consumption, not past average consumption.  

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## 1 answers

### Answer by username_2 on 2017-02-21. Score: 2

Consider the risk free steady state. In that situation the Euler equation becomes:

$$ \left(\frac{C\_{t+1}-b \overline{C\_{t}}}{C\_t-b \overline{C\_{t-1}}}\right)^{-\sigma}=\frac{1}{\beta \cdot (1+r)}$$

Which implies

$$ \Rightarrow \ln[C\_{t+1}-b \overline{C\_{t}}] -\ln[C\_t-b \overline{C\_{t-1}}] = \frac{\ln[\beta] + \ln[1+r]}{\sigma}$$

Assume that in the non-stochastic setting that growth is constant of the form:
$$ C\_{t+1} = \gamma \cdot C\_{t}$$

Note that by definition
$$ \overline{C\_{t}} \equiv \frac{1}{t}\sum^t\_{\tau=1} C\_{\tau}$$
Under a constant growth rate
$$ \Rightarrow \overline{C\_{t}}=\frac{1}{t}\sum^t\_{\tau=1} \gamma^{\tau-t}C\_{t} = \frac{C\_t}{t} (1+\frac{1}{\gamma-1}) = \frac{C\_t}{t} (\frac{\gamma}{\gamma-1})$$
$$ \Rightarrow \overline{C\_{t+1}}= \frac{C\_{t+1}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \frac{C\_{t}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \overline{C\_{t}}$$

$$ \Rightarrow \frac{\ln[\beta] + \ln[1+r]}{\sigma} = \ln[C\_{t+1}-b \overline{C\_{t}}] -\ln[C\_t-b \overline{C\_{t-1}}] $$
$$ = \ln[\gamma C\_{t}-b \gamma \overline{C\_{t-1}}] -\ln[C\_t-b \overline{C\_{t-1}}] = \ln[\gamma]$$

Therefore the one period growth rate in consumption ($\gamma$) in the non-stochastic model is $$ \gamma = e^{\frac{\ln[\beta] + \ln[1+r]}{\sigma}} = [\beta(1+r)]^\sigma$$


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