WARNING: This post contains an excessive amount of math. If you find math unbearable and/or difficult to understand, do not attempt to read this.
A little bit ago, I decided to combine a New Keynesian model with Rotemberg style pricing and a Cash-Credit goods model. Here is a derivation of that model:
Households
Households maximize
U=E0∞∑t=0βt(θlogc1t+(1−θ)logc2t−γlognt)
subject to
Mt−1+Rt−1Bt−1+Wtnt=PtCt+Bt+Mt+Ptτt
Mt≥Ptc1t
Ct=c1t+c2t
Where
c1t is the part of the consumption good that the household buys in the cash market,
c2t is the part of the consumption good that the household buys in the credit market,
Ct is total spending on the consumption good,
Wt is the nominal wage rate,
nt is hours worked by the household,
Mt is the nominal money supply,
Bt is the supply of government bonds,
Pt is the price of the consumption good, and
τt is lumps sum taxes/transfers from the government.
The households maximization problem can be written as
L=U+λ0t(Mt−1+Rt−1Bt−1+Wtnt−PtCt−Bt−Mt−Ptτt)+λ1t(Mt−Ptc1t)+λ2t(Ct−c1t−c2t)
Solving the Lagrangian gives the following First Order Conditions:
(1)1c2t=βEt1c2t+1RtEtPtPt+1
(2)WtPt=γ1−θc2tnt
(3)θc1t=1−θc2t(2−1Rt)
(4)Mt=Ptc1t
Retail Firms
Retail firms maximize profits,
PtYt−∫10Pt(i)yt(i)di subject to the production technology
Yt=[∫10yt(i)ϵ−1ϵdi]ϵϵ−1.
Substituting the production technology into the profit function yields
Pt[∫10yt(i)ϵ−1ϵdi]ϵϵ−1−∫10Pt(i)yt(i)di
Taking the derivative of this with respect to
yt(i) gives the retail firm's first order condition:
yt(i)=Yt(Pt(i)Pt)−ϵ
Since the retail firm is perfectly competitive, its profits are equal to zero. We can therefore set profit equal to zero and plug in the first order condition to get the definition of the price level
P1−ϵt=∫10Pt(i)1−ϵ
Wholesale Firms
There is a continuum of monopolistically competitive wholesale firm who are subject to the quadratic price adjustment cost
φ2(Pt(i)Pt−1(i)−1)2Yt
First, each wholesale firm minimizes total costs,
WtPtnt(i) subject to the production function
yt(i)=atnt(i). This problem can be set up as
L=−WtPtnt+mct(atnt(i)−yt(i))
which yields
(5)WtPt=mctnt(i)
The Lagrangian multiplier in this problem is the marginal cost of production (hence the name
mct).
Each retail firm now maximizes the expected sum of all future profits which is discounted by the 'stochastic discount factor' with the real interest rate replacing the time preference rate and is subject to the retail firm's demand function,
yt(i)=Yt(Pt(i)Pt(i))−ϵ. Since the maximization problem for this is so obscenely long, I won't write it down, I'll just skip to the first order condition.
0=(1−ϵ)YtPt+ϵmctYtPt(i)−φ(Pt(i)Pt−1(i)−1)YtPt−1(i)+βEt(c2t+1c2t)−1φ(Pt+1(i)Pt(i)−1)Pt+1(i)YtPt(i)2
Consider the fact that, since each firm has the same level of technology, the same demand curve, and the price adjustment costs, every firm chooses the same price. Given this as well as the fact that the rate of inflation,
πt is equal to
PtPt−1, the 'New Keynesian Phillips Curve' above can be written as
(6)0=(1−ϵ)+ϵmct−φπt(1+πt)+βEt(c2t+1c2t)−1φπt+1(1+πt+1)Yt+1Yt
Equilibrium
Equations 1-6 can be combined with a description of government policy to complete this model. The money supply and the wage have been rewritten in real terms.
(1)1c2t=βEt1c2t+1Rt1+πt+1
(2)wt=γ1−θc2tnt
(3)θc1t=1−θc2t(2−1Rt)
(4)mt=c1t
(5)wt=mctnt
(6)0=(1−ϵ)+ϵmct−φπt(1+πt)+βEt(c2t+1c2t)−1φπt+1(1+πt+1)Yt+1Yt
(7)logRt=β−1β+ϕππt+υt
(8)logat=ρlogat−1+εat
(9)Yt=Ct+φπ2tYt
(10)Ct=c1t+c2t
(11)υt=ρυt−1+εit;
Impulse Response Functions
Here is the impulse response function (in log deviations from steady state) for the technology shock,
εat where
V is the velocity of money:
And here is the impulse response function for the monetart policy shock,
εit: