<p>In the presentation that the OP linked to, "compactness" is not assumed, but derived as a result from more primitive properties that are assumed, one of which is the "twice differentiability" of the functions involved.</p>

<p>The presentation deals with the vary basic deterministic growth model in discrete time. The per capita production function in each period is $g(k_t)$ for which it is assumed that it is continuous and differentiable $g'&gt;0,\;\; g''&lt;0$. The law of motion of capital is then</p>

<p>$$k_{t+1} = g(k_t) + (1-\delta)k_t -c_t$$</p>

<p>with $0&lt;\delta&lt;1$ being the depreciation rate. The sequence starts at some arbitrary, finite initial value.</p>

<p>The presentation wants to show the existence and uniqueness of the solution. To do that, in page 6 it first proves that the "choice set" is bounded -namely that the set from which the economic agents may choose the optimal levels of consumption and capital is <em>not</em> the whole $\mathbb R_+$.</p>

<p>This is proved by considering the <em>maximal</em> sequence of capital levels, irrespective of utility optimization considerations. If the maximal sequence is bounded, it means that the optimal sequence can only be chosen from a bounded set of values.  </p>

<p>The maximum capital sequence obtains if we set consumption equal to zero: what happens when we consume nothing and we use all our capital stock <em>and</em> all the new output as capital only.
We have then the sequence</p>

<p>$$k^{max}_{t+1} = g(k^{max}_{t}) + (1-\delta)k^{max}_{t}$$</p>

<p>Now that consumption is set to zero, the sequence of capital is governed by a non-linear first-order difference equation. A sufficient (although not necessary) condition for the maximal sequence to be bounded is that this difference equation is asymptotically globally stable, namely, if starting from any initial point we converge to a fixed point (it is not necessary because we could have bounded oscillations also and obtain a bounded sequence).  </p>

<p>The fixed point of the maximal capital sequence satisfies</p>

<p>$$k^{max}_*:  k^{max}_{t+1}=k^{max}_{t} \implies g(k^{max}_*) = \delta k^{max}_* $$ </p>

<p>By standard theory, global stability of the fixed point will obtain if the derivative of the difference equation, evaluated at the fixed point, is less than unity in absolute value,</p>

<p>$$\left |\frac {\partial k^{max}_{t+1}(k^{max}_*)}{\partial k^{max}_{t}}\right| = g'(k^{max}_*)+1-\delta &lt;1 \implies g'(k^{max}_*) &lt; \delta$$</p>

<p>At the fixed point, it holds that </p>

<p>$$\frac {g(k^{max}_*)}{k^{max}_*} = \delta $$</p>

<p>For fixed point stability we want </p>

<p>$$g'(k^{max}_*) &lt; \delta$$</p>

<p>So fixed point stability requires</p>

<p>$$g'(k^{max}_*)&lt;\frac {g(k^{max}_*)}{k^{max}_*}$$</p>

<p>or that "the marginal product will be less than the average product". Under the assumption that $g''&lt;0$ everywhere, this will also hold for the fixed point. Hence, we obtain fixed point stability, and so also convergence and so boundedness of the maximal capital sequence. This tells us why we assume 2nd-order differentiability: by having the 2nd derivative and making assumptions about it, we can easily determine whether the equilibrium points are unique and things like that. Note also that differentiability of any degree implies continuity.</p>

<p>If the maximal capital sequence is bounded, so will be any other capital sequence, and hence the choice set for capital is bounded. If it is for capital it is also for consumption. Then, the full capital equation expressed as an inequality </p>

<p>$$k_{t+1} \leq g(k_t) + (1-\delta)k_t -c_t$$</p>

<p>defines a set (a unidimensional set, an interval on the real line) that is a) bounded below at zero (by the assumption that capital cannot be negative  b) bounded above as proven c) it can attain its boundary points. So it is closed and bounded and therefore <em>compact</em>.  </p>

<p>More over, we assume that there are no "holes" in it, that there are not "forbidden" values in this set. This makes the set convex.</p>

<p>Given these properties (incl. continuity of the production function), by standard theorems we are assured that a maximum, i.e. a solution exists, and it is unique, which is what the presentation wanted to prove at this point.</p>

CC BY-SA 4.0
2018-08-24T09:10:27.150Z
2018-08-24T12:27:03.267Z
2018-08-24T12:27:03.267Z
Alecos Papadopoulos
<p>Field of Economics became mathematical and model based. I can grasp the appeal of the model but fail to understand why do we introduce certain assumptions.</p>

<p>(Actually I am trying to relax the assumptions of the model, Ramsey Cass Kooperman (RCK) model, just as an exercise but before I can relax them I need to know why they were significant in the first place)</p>

<p>So, in the context of RCK model
Why does the assumption of </p>

<p>1.Compactness, </p>

<p>2.twice continuously differentiable function, and </p>

<ol start="3">
<li>homogeneity (homothetic)  are taken. </li>
</ol>

<p>Why are the relevant?</p>

<p>I know this that homogeneity produce constant returns to scale with Cobb Douglas production function and lt also implies homothetic. Why do we need homothetic functions? </p>

CC BY-SA 4.0
2018-07-24T13:25:52.013Z
Hi: I don't know the other ones but twice differentiable implies that, if the hessian of the function is psd, then the function is convex. The convexity of a function implies that the optimization of the function will lead to a global maximum ( minimum ) which is helpful when optimizing it. Also, if a function is convex, then the algorithms for optimizing it are  more straightforward.
mark leeds
CC BY-SA 4.0
2018-07-24T19:36:06.123Z
just read it again and what I said is related to the concavity of profuction function because a function is convex if and only if the negative of it is concave. So, if you have a concave function, you can multiply by -1 to make it convex.
mark leeds
CC BY-SA 4.0
2018-07-24T23:02:37.777Z
your welcome: I'd be interested in reasons for compactness ( I think it has to do with implying boundedness of the function but you may need other conditions also ) and constant elasticity of substitution so, if you get any answers for those outside of this thread and can post them here, it's appreciated.
mark leeds
CC BY-SA 4.0
2018-07-25T05:51:18.180Z
I'll definitely post them here!
Elina Gilbert
CC BY-SA 4.0
2018-08-21T09:51:05.410Z
Τhis post asks for a compact introduction to almost all the underlying mathematical theory of core micro- and macro-economics, and their logic (for the supplier side). And then, the "what happens if they are relaxed" sends us to explore Space... this is too broad to be answerable, please try to considerably narrow it down. Your 2nd post about the Euler things was much more suitable than this one, so you can do it.
Alecos Papadopoulos
CC BY-SA 4.0
2018-08-21T18:31:37.670Z
@Alecos Papadopoulos - Thanks for suggestion. I have completely reframed the question. Hope it's clear now.
Elina Gilbert
CC BY-SA 4.0
2018-08-21T18:35:59.167Z
(+1) for the effort. Now, clarifications: "Compactness" is usually a property of a set. Which set are we talking about here?
Alecos Papadopoulos
CC BY-SA 4.0
2018-08-21T19:38:20.417Z
What I have learnt from reading Compactness means that subsequence of a series is convergent. And here series are dated consumption.
Elina Gilbert
CC BY-SA 4.0
2018-08-21T23:37:58.837Z
That's a confusing way to talk about compactness. And the consumption series is not in general convergent, it lives in the real numbers so we can form a sequence that is not convergent (also, remember that "non-convergence" does not necessarily mean "going to infinity", it may also mean a cyclical behavior). The optimal consumption sequence may be convergent, but that's another matter. Can you provide a link or a reference where these claims appear?
Alecos Papadopoulos
CC BY-SA 4.0
2018-08-22T16:56:22.893Z
https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://homepage.ntu.edu.tw/~yitingli/file/macro%2520and%2520money/Cass_Koopmans_2011.pdf&ved=2ahUKEwiItfeWkIHdAhUGNo8KHZKvAk4QFjAHegQIABAB&usg=AOvVaw3UaUczF4B5cgMOzlA4wvo8
Elina Gilbert
CC BY-SA 4.0
2018-08-22T16:58:06.320Z
I am studying the model from various text but above link provides a nice summary of the model and specify use of the assumption of Compactness, but just like books it doesn't provide explanation for bringing this into picture.
Elina Gilbert
<macroeconomics>
Assumptions of Ramsey Cass Model
Elina Gilbert
# Assumptions of Ramsey Cass Model

(Asked by: username_0 on 2018-07-23)

Field of Economics became mathematical and model based. I can grasp the appeal of the model but fail to understand why do we introduce certain assumptions.

(Actually I am trying to relax the assumptions of the model, Ramsey Cass Kooperman (RCK) model, just as an exercise but before I can relax them I need to know why they were significant in the first place)

So, in the context of RCK model
Why does the assumption of

1.Compactness,

2.twice continuously differentiable function, and

3. homogeneity (homothetic) are taken.

Why are the relevant?

I know this that homogeneity produce constant returns to scale with Cobb Douglas production function and lt also implies homothetic. Why do we need homothetic functions?

**Tags:** macroeconomics

> **Comments:**
> **username_1**: Hi: I don't know the other ones but twice differentiable implies that, if the hessian of the function is psd, then the function is convex. The convexity of a function implies that the optimization of the function will lead to a global maximum ( minimum ) which is helpful when optimizing it. Also, if a function is convex, then the algorithms for optimizing it are  more straightforward.  
> **username_1**: just read it again and what I said is related to the concavity of profuction function because a function is convex if and only if the negative of it is concave. So, if you have a concave function, you can multiply by -1 to make it convex.  
> **username_1**: your welcome: I'd be interested in reasons for compactness ( I think it has to do with implying boundedness of the function but you may need other conditions also ) and constant elasticity of substitution so, if you get any answers for those outside of this thread and can post them here, it's appreciated.  
> **username_0**: I'll definitely post them here!  
> **username_2**: Τhis post asks for a compact introduction to almost all the underlying mathematical theory of core micro- and macro-economics, and their logic (for the supplier side). And then, the "what happens if they are relaxed" sends us to explore Space... this is too broad to be answerable, please try to considerably narrow it down. Your 2nd post about the Euler things was much more suitable than this one, so you can do it.  
> **username_0**: @username_2 - Thanks for suggestion. I have completely reframed the question. Hope it's clear now.  
> **username_2**: (+1) for the effort. Now, clarifications: "Compactness" is usually a property of a set. Which set are we talking about here?  
> **username_0**: What I have learnt from reading Compactness means that subsequence of a series is convergent. And here series are dated consumption.  
> **username_2**: That's a confusing way to talk about compactness. And the consumption series is not in general convergent, it lives in the real numbers so we can form a sequence that is not convergent (also, remember that "non-convergence" does not necessarily mean "going to infinity", it may also mean a cyclical behavior). The optimal consumption sequence may be convergent, but that's another matter. Can you provide a link or a reference where these claims appear?  
> **username_0**: https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://homepage.ntu.edu.tw/~yitingli/file/macro%2520and%2520money/Cass_Koopmans_2011.pdf&ved=2ahUKEwiItfeWkIHdAhUGNo8KHZKvAk4QFjAHegQIABAB&usg=AOvVaw3UaUczF4B5cgMOzlA4wvo8  
> **username_0**: I am studying the model from various text but above link provides a nice summary of the model and specify use of the assumption of Compactness, but just like books it doesn't provide explanation for bringing this into picture.  

---

## 1 answers

### Answer by username_2 on 2018-08-24. Score: 2 | Accepted answer

In the presentation that the OP linked to, "compactness" is not assumed, but derived as a result from more primitive properties that are assumed, one of which is the "twice differentiability" of the functions involved.

The presentation deals with the vary basic deterministic growth model in discrete time. The per capita production function in each period is $g(k\_t)$ for which it is assumed that it is continuous and differentiable $g'>0,\;\; g''<0$. The law of motion of capital is then

$$k\_{t+1} = g(k\_t) + (1-\delta)k\_t -c\_t$$

with $0<\delta<1$ being the depreciation rate. The sequence starts at some arbitrary, finite initial value.

The presentation wants to show the existence and uniqueness of the solution. To do that, in page 6 it first proves that the "choice set" is bounded -namely that the set from which the economic agents may choose the optimal levels of consumption and capital is *not* the whole $\mathbb R\_+$.

This is proved by considering the *maximal* sequence of capital levels, irrespective of utility optimization considerations. If the maximal sequence is bounded, it means that the optimal sequence can only be chosen from a bounded set of values.

The maximum capital sequence obtains if we set consumption equal to zero: what happens when we consume nothing and we use all our capital stock *and* all the new output as capital only.
We have then the sequence

$$k^{max}\_{t+1} = g(k^{max}\_{t}) + (1-\delta)k^{max}\_{t}$$

Now that consumption is set to zero, the sequence of capital is governed by a non-linear first-order difference equation. A sufficient (although not necessary) condition for the maximal sequence to be bounded is that this difference equation is asymptotically globally stable, namely, if starting from any initial point we converge to a fixed point (it is not necessary because we could have bounded oscillations also and obtain a bounded sequence).

The fixed point of the maximal capital sequence satisfies

$$k^{max}\_\*: k^{max}\_{t+1}=k^{max}\_{t} \implies g(k^{max}\_\*) = \delta k^{max}\_\* $$

By standard theory, global stability of the fixed point will obtain if the derivative of the difference equation, evaluated at the fixed point, is less than unity in absolute value,

$$\left |\frac {\partial k^{max}\_{t+1}(k^{max}\_\*)}{\partial k^{max}\_{t}}\right| = g'(k^{max}\_\*)+1-\delta <1 \implies g'(k^{max}\_\*) < \delta$$

At the fixed point, it holds that

$$\frac {g(k^{max}\_\*)}{k^{max}\_\*} = \delta $$

For fixed point stability we want

$$g'(k^{max}\_\*) < \delta$$

So fixed point stability requires

$$g'(k^{max}\_\*)<\frac {g(k^{max}\_\*)}{k^{max}\_\*}$$

or that "the marginal product will be less than the average product". Under the assumption that $g''<0$ everywhere, this will also hold for the fixed point. Hence, we obtain fixed point stability, and so also convergence and so boundedness of the maximal capital sequence. This tells us why we assume 2nd-order differentiability: by having the 2nd derivative and making assumptions about it, we can easily determine whether the equilibrium points are unique and things like that. Note also that differentiability of any degree implies continuity.

If the maximal capital sequence is bounded, so will be any other capital sequence, and hence the choice set for capital is bounded. If it is for capital it is also for consumption. Then, the full capital equation expressed as an inequality

$$k\_{t+1} \leq g(k\_t) + (1-\delta)k\_t -c\_t$$

defines a set (a unidimensional set, an interval on the real line) that is a) bounded below at zero (by the assumption that capital cannot be negative b) bounded above as proven c) it can attain its boundary points. So it is closed and bounded and therefore *compact*.

More over, we assume that there are no "holes" in it, that there are not "forbidden" values in this set. This makes the set convex.

Given these properties (incl. continuity of the production function), by standard theorems we are assured that a maximum, i.e. a solution exists, and it is unique, which is what the presentation wanted to prove at this point.


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