Microeconomic Foundations

Scarth Chapter 1

Makroekonomi 2, S1, FEUI, 2009 – Arianto A. Patunru

Firm

Firms’ Problem

• Y = F (N, K)

• FN, FK > 0; FNN, FKK < 0; FNK = FKN > 0

• Max PV = ∑(1/(1+r))t[PF(Nt,Kt) - WNt

- PI It - b PI It2]; b > 0

• s.t. It = (Kt+1 - Kt) + δKt

Decision Rules

• Assm static expectation

• ∂PV/∂Nt = (1/(1+r))t(PFN - W) = 0 (1)

• ∂PV/∂Kt = (1/(1+r))t[PFK - PI(1- δ)

+ 2bPI(1- δ)It]

+ (1/(1+r))t-1[- PI - 2bPIIt-1] = 0 (2)

Simplification on (1)

• Assm CRS

• Eg Cobb-Douglas

• F(N,K) = KαN1-α, 0<α<1 (3)

• FN = (1- α)(N/K)-α, FK = α(N/K)1-α

• FK = α[W/P(1- α)](α - 1)/α by (1) and (3)

• Since W/P and α constant, FK constant

Simplification on (2)

• Assm PI = P

• Mult by (1+r)t

• Sub in: B = [FK - (r + δ)]/2b (4)

• Then (2) becomes

It – ((1+r)/(1- δ))It-1 + B/(1- δ) = 0

Simplification on (2) … cont’d

• Evaluated at eq:

(It - I*) = ((1+r)/(1- δ))(It-1 - I*)I* = δK* = B/(r + δ) (4)

• Assm r > 0, δ > 1• So ((1+r)/(1- δ)) > 1• And It can be +∞, I*, or -∞• By (3) and (4): I = 1/2b ((FK/(r + δ)) - 1)• Consequence: K = d(K* - K)

Implication

• Invest when FK > (r + δ)

• Set net inv equal to a fraction of gap between desired and actual capital

• Set gross inv equal to optimal replacement investment

• Set PV of income equal to market value of equities (Tobin’s q)

Other important points

• Inv must be higher at higher levels of output

• Proof. Tot-diff on

Y = F(N,K) and I = 1/2b ((FK/(r + δ)) – 1

• Set dK = 0, get dY = FNdN and

dI = [FKN/(2b(r+ δ)FN)]dY - [FK/(2b(r+ δ)2]dr

• Both terms in brackets (+), so:

Other important points… cont’d

• IY = FKN/(2b(r+ δ)FN) > 0

• Ir = -FK/(2b(r+ δ)2) < 0

Household

Household’s problem

• U = F(C)

• F’>0, F’’<0

• Max PV = ∑(1/(1+ρ))tCt

• s.t. Ct = Yd - (At+1 - At) - h(At/Yd)

h’<0, h’’>0

At = qtVt + Mt/Pt recall q fr Tobin

HH constraint (1)

Other financing constraints

• qt(Vt+1 - Vt) = Kt+1 - Kt Firm’s (2)

• PtGt = Mt+1 - Mt Govt’s (3)

(no taxes, no bonds, only money issuance)

• Ydt = Ct + At+1 - At

(disp inc = cons, sav, cap gains) (4)

Redefining Yd

• Ydt = Ct + qt(Vt+1 - Vt) + Vt (qt+1 - qt)

+ (1/Pt)(Mt+1 - Mt) - (Mt/Pt)((Pt+1- Pt)/Pt)• By (1)-(4) and It = (Kt+1 - Kt) + δKt

• Ydt = Ct + It + δKt + Vt (qt+1 - qt)

+ Gt - (Mt/Pt)((Pt+1- Pt)/Pt)• Subs in Yt = Ct + It + Gt:• Yd

t = Yt - δKt + Vt (qt+1 - qt) - (Mt/Pt)((Pt+1- Pt)/Pt)• If static exp and const exp infl:

Yd = Y - δK - (M/P)π

Decision rule: saving

• ∂PV/∂At = (1/(1+ ρ))t[(-h’(At/Yd)/Yd)+1]

- (1/(1+ ρ))t-1 =0

• Mult by (1+ ρ)t -h’(At/Yd) = ρYd

• Impl: At+1 = At (due to const ρ & exp Yd )

• So: C = Yd - h(A/Yd) ∂C/∂Yd = 1 + h’AY/Yd2 ie a fraction Pigou effect ∂C/∂A = - h’/Yd > 0

Decision rule: portfolio

• (M/P) + (qV) = (M/P)D + (qV)D = A

• (M/P)D - (M/P) = (qV) - (qV)D Walras

• Money demand fn L(Y,i,A)

• Equity demand fn V(Y,i,A)

Ly > 0 , Li < 0

Vy < 0 , Vi > 0

LA + VA = 1

Labor Market

Wage setting

• Recall FN = W/P• Two costs of money wage-setting: (i) cost

from wage deviation from its eq level, (ii) cost from renegotiation

• Opt rate of wage change is obtained by:• Min PV = ∑(1/(1+r))t{(wt - ŵt)2 +

β[(wt - ŵt) - (wt-1 - ŵt-1)]2}w = lnW, β is adj cost of deviation (or taste/tech parameter)

Decision rule

• Set ∂PV/∂wt = 0

• Result: wt+1 - ŵt+1 = γ(wt-1 - ŵt-1) , 0< γ <1

γ is the char root of SOC

• Rewrite: wt+1 - wt = ŵt+1 - ŵt + (1-γ)(ŵt - wt)

Wage-setting to price-setting (1)

• Recall EAPC: P˜/P = f.((Y-Ŷ)/Ŷ) + π

• In log: p = f.(y - ŷ) + π (1)

• Rewrite wage rule: w = ŵ˜ + a(ŵ - w), a>0

(to imply: wage is sticky)

• Recall prod fn Y = KαN1-α, labor demand fn W/P = FN = (1- α)(Y/N), and log-lin version of agg demand y = φg + θ(m-p) +ψp

Wage-setting to price-setting (2)

• Wage level that makes n = ň is thus

ŵ - p = ln(1 – α) + ŷ – ň

• Combined with labor demand fn:

(w - ŵ) = (p - p) + (y - ŷ) - (n - ň)

• And ŵ˜ = p