Question

Start with the demand side. The household in question has the following Cobb Douglas utility function: The household also faces the following budget constraint: The above says that the household's after-tax income, (1-r)V,. is divided between consumption of goods and services, C, and the amount spent on housing services, (r+0+%)p"H, . This latter variable can be thought of as the user cost of housing and consist of the rate of interest (r), the rate of depreciation () and residential taxes (r). Saving is zero and you can think of this as representing a young household with no assets that finances the purchase of the house with 100 per cent debt. a) Using equations (1) and (2) to form the Lagrangian, derive an equation for the desired demand for housing (H). Find as well the equilibrium price of a house

Answer 1

According to the question:

Maximize the utility function subject to the given constraint. We use lagrangian to solve for H*

$\textit{L} = H_t^{\eta} C_t^{1-\eta} - \lambda((1-\tau)Y_t - C_t- (r+\delta+\tau_R) p_t^{H} H_t)$

$\frac{dL}{dH} = \eta H_t^{\eta -1} C_t^{1-\eta} +\lambda (r+\delta+\tau_R)p_t^{H} =0$

$\frac{dL}{dC} = (1-\eta) C_t^{-\eta } H_t^{\eta} +\lambda =0$

$\frac{dL}{d\lambda} = (1-\tau) Y_t - C_t -(r+\delta+\tau_R)p_t^{H} H_t=0$

Using the equations dL/dH and dL/dC , we get

$\eta H_t^{\eta-1} C_t^{1-\eta} =- \lambda(r+\delta+\tau_R)p_t^{H}$

$(1-\eta) C_t^{-\eta} H_t^{\eta} = - \lambda$

Using the above equations we get the relation between H and C:

$\eta H_t^{\eta-1} C_t^{1-\eta} = H_t^{\eta} (1-\eta)C_t^{\eta}(r+\delta+\tau_R)p_t^{H}$

$H_t=\frac{\eta}{(r+\delta+\tau_R)p_t^{H} } C_t$

Substituting the above value in the equation of $\frac{dL}{d\lambda}$

$(1-\tau) Y_t = C_t +(r+\delta+\tau_R)p_t^{H} H_t$

$(1-\tau) Y_t = C_t + (r+\delta+\tau_R)p_t^{H} (\frac{\eta}{(r+\delta+\tau_R)p_t^{H}}) C_t$

$(1-\tau) Y_t = C_t + \eta C_t$

$\frac{(1-\tau)}{1+\eta} Y_t = C_t$

So, the value of H:

$H_t=\frac{\eta(1-\tau)}{(1+\eta)(r+\delta+\tau_R)p_t^{H} } Y_t$

Equilibrium price p_H:

$MRS_{HC} = \frac{p_H}{p_C}$

$\frac{\eta H_t^{\eta-1} C_t^{1-\eta}}{1-\eta H_t^{\eta} C_t^{-\eta}} = \frac{p_H}{p_C}$

we know p_c= 1

$\frac{\eta C_t}{1-\eta H_t} = p_H$