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(ct, Mt/Pt) = ctσ(κ Mt/Pt)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
Ptct + Mt + Bt + Stat = Yt + Mt-1 + (1 + it-1)Bt-1 + (St + Dt)at-1,
where it denotes the nominal interest on bonds held between period t and t + 1 (and hence it-1 on bonds held between t - 1 and t).
(a): Let φ(ct, it) denote the real money demand function. Using the first-order conditions of the representative consumer's Lagrangian, generate the function φ(ct, it) (i.e., solve for real money demand (Mt/Pt) as a function of ct and it).
(b): What is the elasticity of real money demand with respect to it/(1 + it)?
(c): What is the elasticity of real money demand with respect to ct?
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...