How to relate real rate of return on capital to bond interest rate: Lagrangian

Question

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?

Answer 1

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 $t+1$.