Monetary Demand and Supply, Leisure

We assume that the person's money demand behavior satisfies following statement

$\frac{M_t}{P_t} = L(y_t,R_t)$

where M means demand of nominal money, P means price index, R means prevailing rate of interest on some relevant assets. Variable y makes function L increase and variable R makes function L decrease.

A Formal Model

consider a hypothetical household that seeks at time $t$ to maximize the multiperiod utility function :

$u(c_t,l_t) + \beta u(c_{t+1},l_{t+1}) + \beta u(c_{t+2},l_{t+2}) + ...$

here the $c_t$ and $l_t$ are the household's consumption of goods and leisure, respectively during the period $t$.

Given above assumptions, the household's budget constraint for period $t$ alone can be written as following statement

$P_ty + M_{t-1} + (1 + R_{t-1})B_{t-1} = P_tc_t + M_t + B_t$

Where B means Bond price. In this assumption, the left side means the total resources available, and the right side means the total expenditure available from the household. To take account of those constraints, first note that the constraint for $t + 1$ can be written as

$B_t = \frac{P_{t+1}(c_{t+1} - y) + M_{t+1} - M_t + B_{t+1}}{1+R_t}$

When the period $t+1$, we can write above equation to following that

$B_{t+1} = \frac{P_{t+2}(c_{t+2} - y) + M_{t+2} - M_{t+1}+ B_{t+2}}{1+R_{t+1}}$

We can eliminate $B_t$ from 3rd equation, and finally arrive at the following equation :

$(1 + R_{t-1})B_{t-1} = [P_t(c_t - y) + (M_t - M_{t-1})] + (1 + R_t)^{-1}[P_{t+1}(c_{t+1} - y) + (M_{t+1} - M_{t})] + (1 + R_t)^{-1}(1 + R_{t+1})^{-1}[P_{t+2}(c_{t+2} - y) + (M_{t+2} - M_{t+1})] + ...$

This single equation then describes the household's entire intertemporal budget constraint, incorporating the constraints for each single period, when the planning horizon is infinite.

A Leisure

A leisure $l_t$ is negatively related to consumption and positively related to real money holdings.

$l_t = \psi (c_t,m_t)$

now we can find maximizing value of first equation of formal model.

$u[c_t,\psi (c_t,\frac{M_t}{P_t})] + \beta u[c_{t+1},\psi (c_{t+1},\frac{M_{t+1}}{P_{t+1}})] + ...$

Lagrangian Expression

To carry out the maximization problem, we can formulate a Lagrangian expression $L_t$ as follows

$L_t = u[c_t,\psi (c_t,\frac{M_t}{P_t})] + \beta u[c_{t+1},\psi (c_{t+1},\frac{M_{t+1}}{P_{t+1}})] + ... + \lambda _{t}[(1+R_{t-1})B_{t-1} - [P_t(c_t - y) + (M_t - M_{t-1})] - (1+R_{t})B_{t} - [P_{t+1}(c_{t+1} - y) + (M_{t+1} - M_{t})] - ...]$

set the resulting expressions equal to zero, we have

$ \frac{\partial L_t}{\partial c_t} = 0$ , $ \frac{\partial L_t}{\partial M_t} = 0$

which mean net marginal utility of consumption increased and net marginal utility of a unit of money holdings additional.

$ S_t $