This is an extract of our Ec210 Key Models & Equations document, which we sell as part of our EC210 - Macroeconomic Problems Notes collection written by the top tier of London School Of Economics students.
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EC210 Key Equations
MICHAELMAS TERM Growth Malthusian Model: Production Function: Y = zF(L,N) Population Growth: N'/N = g(C/N) Market Clearing: C = Y Solow Model: Production Function: Y = zF(K,N) Population Growth: N'/N = (1+n) Market Clearing: C = (1-s)Y Capital Accumulation: Kt+1 = (1-d)Kt + I = (1-d)Kt + sY; k' - k = -dk + sf(k) Solow Model - Labour Augmenting Technological Progress: Production Function: Y = Ka(BN)1-a = zKaN1-a; z = B1-a Capital Accumulation (Per Effective Worker): (1+g+n)(k'e-ke) = sf(ke)-(d+n+g)ke Lucas Critique (Capital flow from rich to poor) Production Function: Y = zKaN1-a Output per Worker: y = Y/N = z(K/N)a = zka; k = (y/z)1/a MPK = azka-1 = az1/ay(a-1)/a (if countries only differ by y, poorer country grows faster) AK Model: If a = 1 Production Function: Y = AK where A = TFP; y = Ak MPK = A (independent of K/N - not diminishing!) growth in SS even if A is constant Capital Accumulation: (1+n)(k' - k)= sAk - (d+n)k If sA > d+n, k growing at constant rate even though A is constant Learning-by-Doing: Production Function: Y = Ka(BN)1-a = Ka(lKN)1-a = l1-aN1-aK where l > 0 indicates positive externality Capital Accumulation: k' - k = sl1-aN1-ak - dk Capital Accumulation Growth: (k'-k)/k = s(lN)1-a - d = x (constant) If x < 0 LR endogenous growth New Implications: Scale effects - removed by modifying so B = lk (instead of lK) Human Capital: Production Function: Y = zuHd (z = marginal product of efficiency units of labour; uHd =
efficiency units of labour) Profit Function: p = Y-wuHd = (z-w)uHd Consumption: c = wuHs (w = real wage; u = fraction of time working; Hs = Stock of human capital) Total Wage Income Human Capital Accumulation: Hs' = b(1-u)Hs where b = efficiency of accumulation Equilibrium: w=z; uHs = uHd - Hs = Hd = H; C = zuH; H' = b(1-u)H Growth Rate of Human Capital: H'/H - 1 = b(1-u) - 1 R&D - One Country Model: Production Function: Y = ALg (Lg = Number of workers engaged in producing output); y = Y/L = A(1-YA) where YA = Fraction of workers engaged in R&D Growth Rate of Knowledge: A = (gA/m)L where m = cost of new inventions If YA is constant, y proportional to A and y = A
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