Question - Money Demand

Consider the infinite-horizon Money in the Utility framework. Assume the representative consumer maximizes lifetime utility by optimally choosing consumption and assets. The period-t utility function is assumed to be:

u(c_t, M_t/P_t) = c_t^σ(κ M_t/P_t)^1-σ,

where the Greek letter "kappa" (κ) in the utility function is a number between zero and one, 0 ≤ κ ≤ 1, over which the representative consumer has no control. The period-t budget constraint of the consumer is

P_tc_t + M_t + B_t + S_ta_t = Y_t + M_t-1 + (1 + i_t-1)B_t-1 + (S_t + D_t)a_t-1,

where i_t denotes the nominal interest on bonds held between period t and t + 1 (and hence i_t-1 on bonds held between t - 1 and t).

(a): Let φ(c_t, i_t) denote the real money demand function. Using the first-order conditions of the representative consumer's Lagrangian, generate the function φ(c_t, i_t) (i.e., solve for real money demand (M_t/P_t) as a function of c_t and i_t).

(b): What is the elasticity of real money demand with respect to i_t/(1 + i_t)?

(c): What is the elasticity of real money demand with respect to c_t?

For full credit, do as follows:

(i) Write out the sequential Lagrangian

(ii) Write out the first order conditions

(iii) Show clearly how you rearrange these first order conditions and which first order conditions are plugged into one another

(iv) Write out the final formula for φ, reducing it as far as possible.

(v) Answer the additional subquestions...