Rental price of capital

我的想法

出租公司？

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Notice that $Y=A{K}^{\alpha }{L}^{1-\alpha }$

$R$ - Rental price of 1 unit of capital, it is a constant

$P$ - The output price, it is a constant

One additional unit of capital is used to produce $MPK$

$MPK=\frac{\delta Y}{\delta K}=\alpha A{K}^{\alpha -1}{L}^{1-\alpha }$

$P\ast MPK$ = What the production firms can earn!

If $P\ast MPK>R$ , a good idea to rent more capital!

If $P\ast MPK<R$ , a good idea to rent fewer capital!

很简单，只要你的产出能够大于你的租金（成本），那就肯定要多产出呗（初级经济学的思想）

In Equilibrium,

$P\ast MPK=R$ < = > $\frac{R}{P}=MPK$

Let us consider the behavior of rental firms

They Benefit of buying c

The Costs of owning/renting out capita

1. A rental firm burrows to buy 1 unit of capital. So, it has to pay interest on loan. Let the price of 1 unit of capital be: $P_k$ Let i be the nominal interest rate
2. When rental firm is renting out capital, the price of capital can change ( i.e. $\Delta P_k$ )

if the price of capital falls, the rental firm losses. i.e. $\Delta P_K<0\to$ A capital loss

if the price of capital increases, the rental firm gains, i.e. $\Delta P_K>0\to$ A capital gain.

3. Let $\delta$ be the rate of depreciation cost, so the depreciation cost becomes: $\delta P_K$

So, 1+2+3 = Total cost of renting out 1 unit of capital = $i\ast P_K-\Delta P_K+\delta P_K$ $=P_K(i-\frac{\Delta P_K}{P_K}+\delta )$

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• Ex 1 A car rental Co buys car for $10,000 each

$i=10\%,\delta =20\%$

Car price rises at 6% per year

$i{P}_K=0.1\ast 10,000=1,000$

$\Delta P_K=0.06\ast 10,000=600$

$\delta P_K=0.2\ast 10,000=2,000$

Total\ cost=1,000-600-2,000=\2,400$

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Total Cost of renting out 1 unit of capital $=P_K(r+\delta )$

$r(realinterestrate)=i-\frac{\Delta P_K}{P_K}$

where $i$ is nominal interest rate, and $\frac{\Delta P_K}{P_K}$ is inflation rate

Then, real cost of renting out 1 unit of capital $=\frac{P_K}{P}(r+\delta )$

Remark: $Realvariable=\frac{Nominalvariable}{Price}$

Q: How does a rental firm decide whether to buy more or less capital stock?

Define real profit per unit of capital :

$ProfitRate=\frac{R}{P}-\frac{P_K}{P}(r+\delta )=MPK-\frac{P_K}{P}(r+\delta )$

Recall: $\frac{R}{P}=MPK$ is the equilibrium condition for the production firms！

if $MPK>\frac{P_K}{P}(r+\delta )$ , Rental firms $\uparrow$ capital stock

if $MPK<\frac{P_K}{P}(r+\delta )$ , Rental firms $\downarrow$ capital stock

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When $Profitrate=0$ ,

$MPK=\frac{P_K}{P}(r+\delta )$

Rental firms should follow this condition to decide the optional amount of $K$ to use!

$Y=A{K}^{\alpha }{L}^{1-\alpha }$ ⇒ $MPK=\frac{\delta Y}{\delta K}=A\alpha K^{\alpha -1}{L}^{1-\alpha }=A\alpha (\frac{L}{K}{)}^{1-\alpha }$

When $MPK=A\alpha (\frac{L}{K}{)}^{1-\alpha }$

and $MPK=\frac{P_K}{P}(r+\delta )$

then $A\alpha (\frac{L}{K}{)}^{1-\alpha }=\frac{P_K}{P}(r+\delta )$

Strongly connected to Solow Model

The left one: if $r$ increases. The right one: When MPK increases.

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Macroeconomics 11 th edition, Mankiw Lecture Notes 3.22

Reference