<p>Ignoring the expectations operator, your lagrangean has two mistakes: first the constraint is per-period so it is also multiplied by the discount factor. Second,  the way you have wrote the time indexes is inconsistent, as regards their interpretation for capital and bonds. If $K_t$ denotes "capital at the beginning of period $t$" (as it does), you should also write $B_{t}$ instead of $B_{t-1}$, to denote bonds held at the beginning of period $t$. This takes with it also the time index on $Q$. </p>

<p>Also  I would advise to write your constraint in "normal form" (as they say in micro), namely with the constraint as "higher than or equal to zero".  </p>

<p>Let's try this, up to a point. </p>

<p>$$\Lambda = \sum_{t=0}^{\infty}\beta^t \Big[U(C_t,L_t) + \lambda_t\big(W_tL_t+R_tK_t+B_{t}+D_t - (P_tC_t + Q_{t+1}B_{t+1}+ P_tI_t)\big)\Big]$$</p>

<p>$$ = \sum_{t=0}^{\infty}\beta^t \Big[U(C_t,L_t)\\ + \lambda_t\big(W_tL_t+R_tK_t+B_{t}+D_t - P_tC_t - Q_{t+1}B_{t+1}- P_t\big[K_{t+1} - (1-\delta)K_t\big]\big)\Big]$$</p>

<p>Then</p>

<p>$$\frac{\partial \Lambda}{\partial K_{t+1}} = 0 \implies -\beta^{t}\lambda_t P_t + \beta^{t+1}\lambda_{t+1}[(1-\delta)P_{t+1} +R_{t+1}] =0$$</p>

<p>and </p>

<p>$$\frac{\partial \Lambda}{\partial B_{t+1}} = 0 \implies -\beta^{t}\lambda_t Q_{t+1} + \beta^{t+1}\lambda_{t+1} =0$$</p>

<p>Equating we get</p>

<p>$$\frac {[(1-\delta)P_{t+1} +R_{t+1}]}{P_t} = \frac 1{Q_{t+1}}$$</p>

<p>Now, $R_{t+1}$ is the <em>nominal</em> gross return to capital, so a $P_{t+1}$ is implicitly in there. If we set $r_{t+1}\equiv R_{t+1}/P_{t+1}$ the relation becomes</p>

<p>$$\frac {[(1+(r_{t+1} -\delta)]P_{t+1}}{P_t} = \frac 1{Q_{t+1}}$$</p>

<p>Taking logs, we get the Fisher equation written for period $t+1$.</p>

CC BY-SA 4.0
2021-02-17T02:31:57.497Z
Just a minor comment: In equilibrium the return on the two assets $B_{t+1}$ and $K_{t+1}$ are identical as a result of the need for rate of change of marginal utilities $\lambda_{t}/\lambda_{t+1}$ to be equal across the two assets.
EndLoop
CC BY-SA 3.0
2015-12-03T15:06:05.007Z
2015-12-03T15:06:05.007Z
Alecos Papadopoulos
<p>Suppose that household resource constrain equation is as follows:
$$P_tC_t + Q_tB_t+ P_tI_t \leq W_tL_t+R_tK_t+B_{t-1}+D_t$$
where $P_t$ is price at time $t$, $Q_t$ is the price of one bond quantity, $B_t$ is bond quantity, $I_t$ is investment, $W_t$ is wage, $L_t$ is labor amount, $R_t$ is nominal capital rental rate, $K_t$ is capital at tie $t$, $D_t$ is dividends.</p>

<p>Taking it into Lagrangian at $t=0$:
$E_0 \sum_{t=0}^{\infty}\beta^t U(C_t,L_t) - \lambda_t[P_tC_t + Q_tB_t+ P_tI_t - (W_tL_t+R_tK_t+B_{t-1}+D_t)]$</p>

<p>Taking partial derivative of lagrangian respect to $K_{t+1}$ where $K_{t+1} = (1-\delta)K_t + I_t$ seems to produce:
$$\frac{\lambda_t}{\lambda_{t+1}} = \frac{R_{t+1}}{P_{t+1}}$$ (Dropped expectation sign, but should be there)</p>

<p>And taking partial derivative of the lagrangian respect to $B_t$:</p>

<p>$$\frac{\lambda_t}{\lambda_{t+1}} = \frac{P_t}{P_{t+1}}\frac{1}{Q_t}$$</p>

<p>Equating these two, </p>

<p>$$R_{t+1} = \frac{P_t}{Q_t}$$</p>

<p>Taking $-\log Q_t = i_t$</p>

<p>$$\hat{R_{t+1}} = \hat{P_t} - \hat{Q_t} = \hat{P_t}+i_t$$
where $\hat{X} = \log X$.</p>

<p>This does not seem to be a right formula to me, and I must have made some mistake. What did I do wrong here? </p>

<macroeconomics><mathematical-economics><interest-rate><capital-returns>
How to relate real rate of return on capital to bond interest rate: Lagrangian
deux
# How to relate real rate of return on capital to bond interest rate: Lagrangian

(Asked by: username_0 on 2015-12-03)

Suppose that household resource constrain equation is as follows:
$$P\_tC\_t + Q\_tB\_t+ P\_tI\_t \leq W\_tL\_t+R\_tK\_t+B\_{t-1}+D\_t$$
where $P\_t$ is price at time $t$, $Q\_t$ is the price of one bond quantity, $B\_t$ is bond quantity, $I\_t$ is investment, $W\_t$ is wage, $L\_t$ is labor amount, $R\_t$ is nominal capital rental rate, $K\_t$ is capital at tie $t$, $D\_t$ is dividends.

Taking it into Lagrangian at $t=0$:
$E\_0 \sum\_{t=0}^{\infty}\beta^t U(C\_t,L\_t) - \lambda\_t[P\_tC\_t + Q\_tB\_t+ P\_tI\_t - (W\_tL\_t+R\_tK\_t+B\_{t-1}+D\_t)]$

Taking partial derivative of lagrangian respect to $K\_{t+1}$ where $K\_{t+1} = (1-\delta)K\_t + I\_t$ seems to produce:
$$\frac{\lambda\_t}{\lambda\_{t+1}} = \frac{R\_{t+1}}{P\_{t+1}}$$ (Dropped expectation sign, but should be there)

And taking partial derivative of the lagrangian respect to $B\_t$:

$$\frac{\lambda\_t}{\lambda\_{t+1}} = \frac{P\_t}{P\_{t+1}}\frac{1}{Q\_t}$$

Equating these two,

$$R\_{t+1} = \frac{P\_t}{Q\_t}$$

Taking $-\log Q\_t = i\_t$

$$\hat{R\_{t+1}} = \hat{P\_t} - \hat{Q\_t} = \hat{P\_t}+i\_t$$
where $\hat{X} = \log X$.

This does not seem to be a right formula to me, and I must have made some mistake. What did I do wrong here?

**Tags:** macroeconomics, mathematical-economics, interest-rate, capital-returns


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## 1 answers

### Answer by username_1 on 2015-12-03. Score: 2

Ignoring the expectations operator, your lagrangean has two mistakes: first the constraint is per-period so it is also multiplied by the discount factor. Second, the way you have wrote the time indexes is inconsistent, as regards their interpretation for capital and bonds. If $K\_t$ denotes "capital at the beginning of period $t$" (as it does), you should also write $B\_{t}$ instead of $B\_{t-1}$, to denote bonds held at the beginning of period $t$. This takes with it also the time index on $Q$.

Also I would advise to write your constraint in "normal form" (as they say in micro), namely with the constraint as "higher than or equal to zero".

Let's try this, up to a point.

$$\Lambda = \sum\_{t=0}^{\infty}\beta^t \Big[U(C\_t,L\_t) + \lambda\_t\big(W\_tL\_t+R\_tK\_t+B\_{t}+D\_t - (P\_tC\_t + Q\_{t+1}B\_{t+1}+ P\_tI\_t)\big)\Big]$$

$$ = \sum\_{t=0}^{\infty}\beta^t \Big[U(C\_t,L\_t)\\ + \lambda\_t\big(W\_tL\_t+R\_tK\_t+B\_{t}+D\_t - P\_tC\_t - Q\_{t+1}B\_{t+1}- P\_t\big[K\_{t+1} - (1-\delta)K\_t\big]\big)\Big]$$

Then

$$\frac{\partial \Lambda}{\partial K\_{t+1}} = 0 \implies -\beta^{t}\lambda\_t P\_t + \beta^{t+1}\lambda\_{t+1}[(1-\delta)P\_{t+1} +R\_{t+1}] =0$$

and

$$\frac{\partial \Lambda}{\partial B\_{t+1}} = 0 \implies -\beta^{t}\lambda\_t Q\_{t+1} + \beta^{t+1}\lambda\_{t+1} =0$$

Equating we get

$$\frac {[(1-\delta)P\_{t+1} +R\_{t+1}]}{P\_t} = \frac 1{Q\_{t+1}}$$

Now, $R\_{t+1}$ is the *nominal* gross return to capital, so a $P\_{t+1}$ is implicitly in there. If we set $r\_{t+1}\equiv R\_{t+1}/P\_{t+1}$ the relation becomes

$$\frac {[(1+(r\_{t+1} -\delta)]P\_{t+1}}{P\_t} = \frac 1{Q\_{t+1}}$$

Taking logs, we get the Fisher equation written for period $t+1$.

> **Comments:**
> **username_2**: Just a minor comment: In equilibrium the return on the two assets $B_{t+1}$ and $K_{t+1}$ are identical as a result of the need for rate of change of marginal utilities $\lambda_{t}/\lambda_{t+1}$ to be equal across the two assets.  

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