<p>Let's think about a unit of non-depreciating capital that has a price of $p_k$, in a static environment where we normalized the price of the consumption good to 1. Furthermore, the unit of capital gives you $r$ units of interest in consumption good units.</p>

<p><strong>tl;dr</strong>: Yes, Because of some equilibrium condition.</p>

<h3>The value of capital</h3>

<p>Let me denote by $\lambda_t$ the value of a marginal consumption good in unit $t$ - this is how much more happy you would be today, if you knew that in time $t$ you get to consume a little bit more.</p>

<p>Then, the utility that a unit of capital generates for you at time $0$ is</p>

<p>$$\lambda_1 r + \lambda_2 r + \lambda_3 r + \dots$$</p>

<p>We assume that capital doesn't give interest in the first period. Note that discount rates are hidden inside $\lambda$; this sequence converges by assumption. This is your discounted real expected return on capital goods.</p>

<h3>Cost of Capital</h3>

<p>Now, what is the opportunity cost of investing into one unit of capital? You forgo $p_k$ units of consumption in period 0.</p>

<h3>Equilibrium Condition</h3>

<p>The cost has to equal the value in equilibrium - the price has to equal the value of capital, otherwise:</p>

<ul>
<li>if the price was higher than the value, people who owned capital would try to sell it</li>
<li>if the price was lower than the value, demand would be larger than supply of capital goods</li>
</ul>

CC BY-SA 3.0
2015-02-16T23:34:32.890Z
2015-02-16T23:34:32.890Z
FooBar
<p>Let $P_k$ be the value of 1 quantity of a capital good. If one does not sell the capital good and keep it, the good provides interests.</p>

<p>In such a case, does standard macro say that all expected discounted real expected returns (or net present value of all future and present returns) be equal to $P_k$?</p>

CC BY-SA 3.0
2015-02-17T15:56:15.580Z
Do you want to assume complete markets? If not, this need not be so.
BKay
<macroeconomics><microeconomics>
Do capital goods have to be valued so that (discounted) real expected return on capital goods equal current present value of capital goods?
cams
# Do capital goods have to be valued so that (discounted) real expected return on capital goods equal current present value of capital goods?

(Asked by: username_0 on 2015-02-16)

Let $P\_k$ be the value of 1 quantity of a capital good. If one does not sell the capital good and keep it, the good provides interests.

In such a case, does standard macro say that all expected discounted real expected returns (or net present value of all future and present returns) be equal to $P\_k$?

**Tags:** macroeconomics, microeconomics

> **Comments:**
> **username_1**: Do you want to assume complete markets? If not, this need not be so.  

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## 1 answers

### Answer by username_2 on 2015-02-16. Score: 0

Let's think about a unit of non-depreciating capital that has a price of $p\_k$, in a static environment where we normalized the price of the consumption good to 1. Furthermore, the unit of capital gives you $r$ units of interest in consumption good units.

**tl;dr**: Yes, Because of some equilibrium condition.

### The value of capital

Let me denote by $\lambda\_t$ the value of a marginal consumption good in unit $t$ - this is how much more happy you would be today, if you knew that in time $t$ you get to consume a little bit more.

Then, the utility that a unit of capital generates for you at time $0$ is

$$\lambda\_1 r + \lambda\_2 r + \lambda\_3 r + \dots$$

We assume that capital doesn't give interest in the first period. Note that discount rates are hidden inside $\lambda$; this sequence converges by assumption. This is your discounted real expected return on capital goods.

### Cost of Capital

Now, what is the opportunity cost of investing into one unit of capital? You forgo $p\_k$ units of consumption in period 0.

### Equilibrium Condition

The cost has to equal the value in equilibrium - the price has to equal the value of capital, otherwise:

* if the price was higher than the value, people who owned capital would try to sell it
* if the price was lower than the value, demand would be larger than supply of capital goods


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