EC9A2 Advanced Macro Analysis - Class #1 Jorge F. Chávez University of Warwick October 29, 2012
Outline 1. Some math 2. Shocking the Solow model 3. The Golden Rule 4. CES production function (more math) 5. A few remarks about PS1
Properties of concave and convex functions Proposition Let f : R 2 R be a C 2 (twice-differentiable) function defined over the open convex set S. Then: f is convex f 11 0, f 22 0 and f 11f 22 (f 12) 2 0 f is concave f 11 0, f 22 0 and f 11f 22 (f 12) 2 0 f 11 > 0 and f 11f 22 (f 12) 2 > 0 = f is strictly convex f 11 < 0 and f 11f 22 (f 12) 2 > 0 = f is strictly concave A corollary of the above proposition is that for a C 2 single-variable real-valued functions (f : R R) to be strictly concave we need f (x) < 0 (sufficient condition). Note that f strictly concave is not a necessary condition for f 11 < 0. Proposition If f is concave, then concavity implies that f is continuous at the interior of its domain. This does not necessarily holds for the boundary points.
Shocking the Solow model Consider the basic discrete-time version of the Solow model as seen in the lecture notes. We start from a steady-state at time 0. Suppose there is a one-time increase in the depreciation rate at the end of time 0 (δ 1 < δ 2 ) How does the economy converges to the new steady-state? How is {y t } t=0 affected? How is the path of log K t affected?
Shocking the Solow model A shock to δ Figure: Solow 1 k t+1 (n+δ2 k t ) f(k t ) (n+δ 1 k t ) f(k) A sf(k t ) sf(k) B k 2 k 1 k t
Shocking the Solow model Some transitions Figure: Some transitions in relevant variables logkt y1 ss Transitory growth y2 ss Kt = k1lt Transition Kt = k2lt t = 0 t t t = 0 t t (a) Path of y t (b) Path of log K t
Shocking the Solow model A shock to s Figure: Solow 1 k t+1 (n+δkt ) f(k t ) s 1 f(k) B s 2 f(k t ) s 1 f(k t ) s 2 f(k) A k 1 k 2 k t
The Golden Rule Consider the basic model in discrete time with population growth. The law of motion of the stock of capital per worker is: k t+1 = sf (k t) + (1 δ) k t 1 + n ϕ (k t ) And the steady state (fixed-point) condition is: sf (k) = (n + δ) k Thus, at a steady-state c = (1 s)f(k) = f(k) (n + δ)k Therefore we can maximize consumption. The condition is: f (k) = (n + δ)
The Golden Rule Figure: The Golden Rule f(k ss ) c 1 (n+δ)k ss s 1 f(k ss ) k 1 k ss
The Golden Rule Figure: The Golden Rule f(k ss ) (n+δ)k ss c 2 s 2 f(k ss ) k 2 k ss
The Golden Rule Figure: The Golden Rule f(k ss ) c 1 (n+δ)k ss c 2 c GR s 1 f(k ss ) s GR f(k GR ) s 2 f(k ss ) k 2 k GR k 1 k ss
CES technology Consider the production function: F (K, L) = [γk (σ 1)/σ + (1 γ) L (σ 1)/σ] σ/(σ 1) We can check that the elasticity of substitution between K and L is constant and equal to σ Recall that the elasticity of substitution is ε KL = d log(k/l) d log(f K /F L ) (the percentage change in relative inputs K/L in response to a percentage change in relative factor prices) For the function above: F K = γk 1/σ F 1/σ F L = (1 γ) L 1/σ F 1/σ Using these results: F K = γ ( ) 1/σ K K ( ) σ F L 1 γ L L = FK 1 γ F L γ In the last expression, think of k = K/L as a function of p = F K /F L. This relationship has the form k = Cp σ for some constant C.
CES as a general case CES approximates a linear production function as σ σ 1 lim = 1 lim F (K, L) = γk + (1 γ) L σ + σ σ + CES approximates a fixed-proportions technology (Leontief) as σ 0. To see this we need to analyze two cases: Suppose K > L. Then: ( ( ) (σ 1)/σ σ/(σ 1) F (K, L) K lim = lim γ + (1 γ)) σ 0 L σ 0 L = lim σ 0 (1 γ) σ/(σ 1) = 1 where the last equality follows because K/L > 1 and (σ 1)/σ. Thus lim F (K, L) = L lim σ 0 σ 0 Suppose K < L. Then, following a similar argument we can establish that lim σ 0 F (K, L) = K. Combining both results: lim F (K, L) = min (K, L) σ 0
CES as a general case CES converges to the Cobb-Douglas function when σ 1. To see this take the logarithm of F and evaluate the limit using L Hospital rule: log ( γk (σ 1)/σ + (1 γ) L (σ 1)/σ) lim log F (K, L) = lim σ 1 σ 1 (σ 1) /σ This implies: = lim σ 1 γk (σ 1)/σ log K/σ 2 (1 γ)l (σ 1)/σ log L/σ 2 γk (σ 1)/σ +(1 γ)l (σ 1)/σ 1/σ 2 = γ log K + (1 γ) log L lim F (K, L) σ 1 = exp (γ log K + (1 γ) log L) = K γ L 1 γ
CES and Inada conditions CES for σ (0, + ) satisfies assumption 2 in the lecture notes: it is strictly increasing and strictly concave in each input. For example, we can check concavity: ( F K (K, L) = γk 1/σ F 1/σ = γ + (1 γ) ( L K ) )(σ 1)/σ 1/σ (σ 1)/σ It is easy to show that F K (K, L) is decreasing in K which implies that F KK < 0. Similarly one can show that F LL < 0. CES for σ = 1 (Cobb-Douglas case) satisfies Inada conditions. For σ 1 CES does not satisfy some parts of the Inada conditions: For σ > 1: For σ < 1: lim F k = +, but, lim F k = γ 1/(σ 1) > 0 K 0 K + lim F k = 0, but, lim F k = γ 1/(σ 1) > 0 K 0 K +
Some variations of the Solow model Consider the Solow model with CES production function given by: F (K t, L t ) = [ γk (σ 1)/σ t ] σ/(σ 1) + (1 γ) L (σ 1)/σ t For the above case, characterize the equilibrium using both competitive markets and a social planner. Add a government: (i) Passive government, (ii) Active government I ll cover some of these in a handout to be posted soon!