McCless]Prof. McCless UCEMA Prof. McCless UCEMA Macroeconomics II Money in the Utility function [ October, 00 Money in the Utility function Money in the Utility function Alternative way to add money to model Utility approximates bene ts from using money in transactions Use sub-utility of the form u(c i t; mi t ; l i t) = u(c i t; mi t ; h i t); Households maximize Households choose sequences of c i t; m i t; k i t+; h i t X E 0 t=0 t u(c i t; mi t ; h i t); subject to the sequence of period t budget constraints, to maximize t=0 c i t + k i t+ + mi t = w t h i t + r t k i t + ( )k i t + mi t + (g t ) M t (g t )M t is a lump-sum transfer of money we use u(c i t; mi t ; m h i t) = ln c i i t + D ln t + Bh i t Household s FOCs
First order conditions for the household are c i t c i t c i t = E t c i t+ P + D t+ m i t = E t c i [r t+ + ( )] t+ B = another equation from teh budget constraint w t c i t + k i t+ + mi t = w t h i t + r t k i t + ( )k i t + mi t + (g t ) M t Prices money Aggregate version of the rst FOC is = E t + D C t C t+ + M t But E t + C t+ = E t C t+ + + D M t+ substituting this into the rst FOC gives C t = E t Repeated substitutions give Prices money The money growth rule is E t C t+ + + D M t+ + D M t = E t C t+ + + E t D M t+ + D M t X = DC t j E t M t+j j=0 M t+ = g t+ M t prices in terms of a sequence of growth rates of money are = DC t M t X j E t j=0 jy g t+k k=
Prices are forward looking depend on the entire expected future growth rates of money We assume money growth follows the process The production sector ln g t = ( ) ln g + ln g t + " g t Production is stard competitive The production function is Y t = t K t H t ; with costs of costs = w t H t + r t K t : Competitive factor market conditions are r t = t K t H t ; w t = ( ) t K t H t : The stochastic shocks for technology, t follow the process Equilibrium conditions ln t = ln t + " t ; Given that there is a unit mass of identical agents, aggregate variables are C t = c i t M t = m i t H t = h i t K t = k i t Stationary states Stationary state version of the FOCs are C = C+ + D M t r = C = w B ( )
The stationary state budget constraint is simply Y = C + K The factor market conditions in a stationary state are r = K H w = ( ) K H The production function is Y = K H Stationary states The rental on capital is r = ( ) From the factor market conditions, get w = ( ) r with this w, get C = w B The rst rst order condition can be written as to give Stationary states = g + D C M=P M=P = D gc g Use the factor market conditions to get H as a function of K, put this into the production function to get r( ) Y = K w Use the stationary state household budget constraint to get K = h r( ) w C i
Use the above equation to get output Using the factor market conditions, get r( ) H = K w Stationary states For the stard economy with = :99, = :0, = :; B = :80; set D = :0 to get same money holdings as in the cia model when g =. Stationary Variable state value r :00 w :0 C :98 K :0 H : Y : :0098g M=P g :99 Log-linear version of the model 0 = e C t + " # g + DC e M=P g E t e C t+ 0 = ew t + re t er t+ [r + ( )] E t ew t+ ; 0 = ew t e Ct ; g E t e + DC fm t ; M=P 0 = C e C t + K e K t+ wh ew t wh e H t rker t [r + ( )] K e K t ; 0 = e Y t e t e K t ( ) e H t ; 0 = er t e t ( ) e K t ( ) e H t ; 0 = ew t e t e K t + e H t ; 0 = f M t eg t f Mt : Log-linear version of the model the three "state" variables, x t = h ekt+ ; f M t ; e i, the ve "jump" variables, y t = her t ; ew t ; C e t ; Y e t ; H e i t h i the stochastic variables z t = et ; eg t
Solve the model as 0 = Ax t + Bx t + Cy t + Dz t ; 0 = E t [F x t+ + Gx t + Hx t + Jy t+ + Ky t + Lz t+ + Mz t ] ; z t+ = Nz t + " t+ ; to get x t+ = P x t + Qz t ; y t = Rx t + Sz t : Linear policy function The solution to the model are linear policy functions x t+ = P x t + Qz t where P = y t = Rx t + Sz t Q = 0 0 0:98 0 0 0: 0 0: 0 0 0:0 :8 ; ; Linear policy function R = S = 0:90 0 0 0: 0 0 0: 0 0 0:00 0 0 0: 0 0 :9 0 0:0 0 0:0 0 :9 0 : 0 : ; Notice the coe cient on prices of a money growth shock Money prices
money prices 0.0 0.08 0.0 0.0 0.0 money 0.0 0.008 prices 0.00 0.00 0.00 0 0 0 0 0 periods Figure : Response of money prices to money growth shock Recall that prices are forward looking in this model The response of money prices to a money growth shock are Real variables a technology shock These are the same as the responses in Cooley-Hansen Seigniorage Government budget constaint is g t = bg t g = M t M t let g be the average government de cit nanced with money issue the rom variable bg t follows the process ln bg t = ln bg t + " g t where " g t N(0; g ), with g as the stard error. De ne ' t as the gross growth rate of money that is required to nance the budget de cit in period t. Then, M t = ' t M t g t = bg t g = (' t ) M t = (' t ) ' t M t
0.0 0.0 Y K w C 0.0 0.00 0 0.00 r H p 0.0 0 0 0 0 80 00 periods Figure : Responses to a :0 impulse in technology Seigniorage Households do not receive direct transfers from government Household budget constraint is Full model is c i t + k i t+ + mi t = w t h i t + r t k i t + ( )k i t + mi t = E t + D ; C t C t+ + M t = E t [r t+ + ( )] ; w t w t+ w t = BC t ; C t + K t+ + M t = w t H t + r t K t + ( )K t + M t ; Stationary states Y t = t Kt Ht ; r t = t Kt Ht ; w t = ( ) t Kt Ht ; bg t g = M t M t : Notice how output changes to cover seigniorage 8
real seigiorage 0.0 0.008 0.00 0.00 0.00 0..... gross stationary state inflation rate Figure : Real seigniorage Annual in ation 0% 0% 00% 00% Corresponding ' :0 :9 : rental 0:0 0:0 0:0 0:0 wages :0 :0 :0 :0 consumption 0:98 0:98 0:98 0:98 real balances =M=P 0:98 0: 0:0 0:008 output : : : : capital :0 :0 :90 :9 hours worked 0: 0:9 0: 0:8 seigniorage = g 0 0:00 0:008 0:0090 Stationary states (Bailey curve) Relation between real seigniorage in ation rate Log-linearization Only two equations need changes 0 = C e C t + K e K t+ + M=P f M t M=P e ' wh ew t wh e H t rker t [r + ( )] K e K t M=P ' f M t the seigniorage equation 0 = geg t + M=P e ' M=P f M t + M=P ' f M t where eg t ln bg t ln = ln bg t is the log of the shock to the government de cit that is nanced by seigniorage. 9
Next slide gives matrices for ' = ' = :9 [cm] cm P = Q = R = S = 0 0 0:98 0 0 0: 0 0 0 0: 0 0:0 0 0:90 0 0 0: 0 0 0: 0 0 0:00 0 0 0: 0 0 :9 0 0:0 0 0:0 0 :9 0 : 0 cm P = Q = R = S = 0:98 0 0 0: 0 :0 0 0: 0:000 0:90 0:8 :8 0: 0:9 0 0 0: 0 0 0: 0 0 0:0 0 0 0:80 0 0 :99 0:00 0: 0:000 0: 0:000 :99 0:00 : 0:00 Response of real variables to a seigniorage shock Response of nominal variables to a technology shock 0
x 0. H r 0. 0 Y 0. w C 0 0 0 0 0 0 Figure : Response to seigniorage shock, ' = :9 0.0 0 0. money 0. 0. prices 0. 0. 0. 0. 0 0 0 0 80 00 periods Figure : Responses of money prices to a technology shock when seigniorage is collected in the stationary state, ' = :9