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Microeconomic Foundations

Makroekonomi 2, S1, FEUI, 2009 – Arianto A. Patunru. Microeconomic Foundations. Scarth Chapter 1. Firm. Firms’ Problem. Y = F (N, K) F N , F K > 0; F NN , F KK < 0; F NK = F KN > 0 Max PV = ∑(1/(1+r)) t [PF( N t ,K t ) - WN t - P I I t - b P I I t 2 ]; b > 0

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Microeconomic Foundations

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  1. Makroekonomi 2, S1, FEUI, 2009 – Arianto A. Patunru Microeconomic Foundations Scarth Chapter 1

  2. Firm

  3. 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

  4. 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)

  5. 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)/αby (1) and (3) • Since W/P and α constant, FK constant

  6. 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

  7. 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)

  8. 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)

  9. 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:

  10. Other important points… cont’d • IY = FKN/(2b(r+ δ)FN) > 0 • Ir = -FK/(2b(r+ δ)2) < 0

  11. Household

  12. 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)

  13. 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)

  14. 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: • Ydt = 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

  15. 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

  16. 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

  17. Labor Market

  18. 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)

  19. 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)

  20. 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

  21. 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

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