Basic Techniques. Ping Wang Department of Economics Washington University in St. Louis. January 2018

Transcription

1 Basic Techniques Ping Wang Department of Economics Washington University in St. Louis January 2018

2 1 A. Overview A formal theory of growth/development requires the following tools: simple algebra simple calculus, particularly the basic concept of differentiation basic diagrammatic analysis, including particularly: o understanding preferences & objectives o understanding technologies & constraints o demand-supply analysis o atemporal constrained optimization o intertemporal constrained optimization While we will overview each of the five diagrammatic tools mentioned above, we will briefly illustrate the usefulness of simple algebra and calculus in three different exercises: understanding the utility function and the production function solving equilibrium employment and real wages decomposing economic growth into productivity and factor accumulation

3 2 B. Quick Review of Preferences and Technologies 1. Preferences At a given point in time, a representative individual allocates her total available time (H) to work (l) and leisure (z), whose preference is represented by a utility function: U = U(c, z), over a composite consumption good (c) and leisure (z = H - l), satisfying: strictly increasing (MU c = U/ c > 0 and MU z = U/ z > 0) diminishing MU ( MU c / c < 0 and MU z / z < 0) diminishing MRS (-dc/dz constant U = MU z /MU c is decreasing in z; IC) complementarity and other regularity conditions (Inada) c c IC E E E IC 0 z z 0 l = H-z H l

4 3 2. Technologies A prototypical one-input individual production function (backyard farming or self-employment Robison Crusoe) is: y = Af(l), A > 0, f satisfying: strictly increasing (MPL = f/ l > 0) diminishing MPL ( MPL/ l < 0) input necessity (f(0) = 0) A prototypical two-input (capital K and labor L) aggregate production function is: Y = AF(K,L), with F satisfying: strictly increasing (MPK= F/ K > 0 and MPL= F/ K > 0) diminishing MPK/MPL ( MPK/ K < 0 and MPL/ L < 0) diminishing MRTS (-dk/dl constant Y is decreasing in L; Isoquants) input necessity (F(0,L) = F(K,0) = 0) complementarity and other regularity conditions (Inada) The scaling factor, A, measures productivity, which is usually referred to as the total factor productivity (TFP). The TFP can change over time both its level and growth rate play crucial roles in growth and development.

5 4 C. Demand-Supply Analysis We will illustrate the demand-supply analysis using the labor market, based on the utility function and one-input production function presented above: U = U(c,z) = U(c,H-l) and y = Af(l) Consider a simple budget constraint (BC): c = y o + w l, where w = real wage rate and y o measures non-labor income. 1. Supply of Labor A rational individual supplies labor when she is indifferent between working and enjoying leisure: MRS = MU z /MU c = w. Under diminishing MRS and assuming normality, the labor supply is upward sloping, as shown in the diagram (higher w raises l). c y o 0 E E IC BC IC w l Note: Normality rules out backward bending labor supply, which is unlikely to a representative individual (an average worker).

6 5 2. Demand for Labor: In a competitive labor market, labor must be hired when its marginal product equals the wage rate prevailed: MPL = f/ l = w. Under diminishing MPL, the labor demand locus is downward sloping. 3. Equilibrium w w 0 E l E l s l d = MPL l Combining labor supply and demand, we obtain equilibrium employment and wage (see point E). c IC IC 4. Comparative-Static Analysis E E BC Consider now non-labor income rises as a y o result of a gift (y o higher). This discourages labor supply, thereby leading to lower employment and higher wage (shift from E to E ). 0 l

7 6 D. Decomposition of Economic Growth Based on the two-input aggregate production function, we can decompose economic growth into productivity and factor accumulation. Such an exercise requires log calculus/total differentiation techniques ( x dx / dt): x dln(x) 1 dx x ln(x y) ln(x) ln(y), ln( ) ln(x) ln(y), y dt x dt x dln(x y) dln(x) dln(y) x y dln(x / y) dln(x) dln(y) x y, dt dt dt x y dt dt dt x y We can now conduct the decomposition exercise: Production function: Y = AF(K,L) = A K α L 1-α Taking natural log: ln(y) = ln(a) + αln(k) + (1-α)ln(L) Y A K L Total differentiating with respect to t: (1 ). Y A K L Thus, economic growth can be decomposed into a TFP component ( A / A), a capital accumulation component ( K / K ) that is weighted by α, and an employment enhancement component ( L / L) that is weighted by 1-α.

8 7 E. Atemporal Optimization (Robison Crusoe) 1. Self-Employment Labor Supply Decision c MPL MRS = ( U/ z)/( U/ c) = w (implicit wage) Af(R) 2. Self-Employment Labor Demand Decision MPL = f/ l = w (implicit wage) 3. Atemporal Self-Employment Equilibrium Goods market equilibrium c IC H R BC c = y = Af (l) (with y o = Af(l) - MPL l) E Af(R) Equilibrium Conditions MRS = w =MPL (points E) y o H R

9 8 F. Intertemporal Optimization: Two-Period Fisherian Model 1. Intertemporal Consumption Choice Lifetime Utility: V = U(c 1, c 2 ), strictly increasing and satisfying diminishing MU, diminishing MRS and other regularity conditions to yield well-behaved intertemporal indifference curves (IIC) a commonly used lifetime utility assumes time-additive periodic utility with constant time discounting: V = u(c 1 ) + β u(c 2 ), with β = 1/(1+ρ) < 1. Lifetime Budget: Since lifetime wealth (Ω) = current income + present value of future income (based on a real rate of interest, r, and the initial real bond holding b 0 ), the intertemporal budget constraint (IBC) is: c 1 + c 2 /(1+r) = b 0 + y 1 + y 2 /(1+r) = Ω 2. Optimized Decision Marginal rate of intertemporal substitution: MRS = ( U/ c 1 )/( U/ c 2 ) = 1 + r

10 9 Diagrammatic illustration: Net Saver in 1 Net Borrower in 1 c 2 (1+r)Ω c 2 (1+r)Ω y 2 A c 2 y 2 E A I IC I BC c 2 E I IC I BC c 1 y 1 Ω c 1 y 1 c 1 Ω c 1 A = autarchy point; E = optimized decision Intertemporal trade triangles => net saving/borrowing decisions Mathematically, we have: o b 0 -(y 2 -c 2 )/(1+r)<0 => net saver in 1 (2 = retirement) o b 0 -(y 2 -c 2 )/(1+r)<0 => net borrower in 1 (1 = childhood/studentship)

11 10 3. From 2-Period to Infinite Horizon To better approximate the real world, we extend the 2-period model to infinite horizon and, for simplicity, allow the representative agent to live forever (finite lives love thy children with positive bequest motives): Lifetime utility: Lifetime budget: t-1 V u(c 1 1 or, ct b0 t1 1 r t1 1 r 2 t-1 1 ) u(c2) u(c3 ) u(ct ) t1 c2 c3 y2 y b0 y r ( 1 r) 1 r ( 1 r) 3 c1, 2 4. From Discrete Time to Continuous Time The continuous-time counterpart is usually more parsimonious: t Lifetime utility: V u(c(t))e dt 0 t-1 Lifetime budget (under constant r): 0 c(t)e dt b0 0 y(t)e dt, or, at each point in time, r y(t) c(t), with Ω(0) = b 0 y t rt rt

12 i Appendix to Basic Techniques Ping Wang Department of Economics Washington University in St. Louis * This note is provided for your bedtime reading to advance better understand of constrained optimization and continuous-time intertemporal optimization.

13 ii A. Constrained Optimization and Lagrangian The constrained optimization problem (COP): Reward function: R(x,y), strictly increasing and strictly concave in each argument and quasi-concave in (x,y), i.e., R x > 0, R y > 0, R xx < 0, R yy < 0, R xx R yy (R xy ) 2 0 Constraint: g(x,y) = z, where the constraint set is convex The Lagrangian Method (λ = Lagrangian multiplier): L(x,y,λ) = R(x,y) + λ [z - g(x,y)] First-order necessary conditions ( L/ x = L/ y = L/ λ = 0): Rx gx 0, R y gy 0, z - g(x,y) = 0 The second-order sufficient conditions are usually met when the reward function is quasi-concave and the constraint set is convex Example: Let the reward function be a standard utility function and the constraint be a standard budget constraint: max U(x,y) subject to p x x + p y y = I Quasi-concavity of U => indifference curves convex toward the origin Then, L = U(x,y) + λ (I - x - py ) and the first-order conditions are: U x λp x = 0, U y λp y = 0, I - p x x - p y y = 0

14 iii B. Intertemporal Optimization in Continuous Time: Optimal Control Optimal control solves continuous-time dynamic constrained optimization problem (DCOP). Rather than using the Lagrangian, we employ current-value Hamiltonian (H) The DCOP with one control (x) and one state (y): max PDV of lifetime reward = R (x, y)e t dt 0 s.t. y g(x, y), y(0) = y 0 > 0 (initial condition) Pontryagin maximum principle (λ = costate): (Step 1) Set up current-value Hamiltonian H(x,y,λ) = R(x,y) + λ g(x,y) (Step 2) Derive first-order necessary condition(s) w.r.t. control(s): R x + λg x = 0 (Step 3) Obtain Euler equation(s) w.r.t. state(s): = ρλ - H/ y = λ(ρ g y )

15 Notes: o The condition, y = H/ λ = g(x,y), is redundant o The sufficient conditions: maximized Hamiltonian H(x*(y),y) is strictly concave in y t a terminal (Transversality) condition is met: lim tye in all models considered, these sufficient conditions are always met; we thus ignore for the sake of brevity Example: Let the reward function be u(c) and g(c,k) = f(k) δk c. (Step 1) Current-value Hamiltonian: H(c,k,λ) = u(c) + λ [f(k) δk c] (Step 2) First-order condition: u c - λ = 0 (Step 3) Euler equation: = λ(ρ + δ f k) iv For multiple n controls and m states, there will be n first-order conditions and m Euler equations

16 v For those interested, it is noted that the Euler equation can be derived using the Fundamental Lemma of Calculus of Variation (CV) True trajectory zt () xt (). () t (Variations) t 0 t 1 time Calculus of variation: minimize deviations from the true trajectory: t R r z z t dt r x x t dt 1 1 ( ) (,, ) (,, ) t0 t0 dr( ) t1 t1 r r 0 dt d t0 x t0 x 0 t dt

17 vi From integration by part: dr( ) t1 t1 r r t1 d r 0 dt dt t t d 0 x x 0 t0 dt x t r d r () tdt 0 x dt x 1 t0 For any (t), the above equation must equal to 0, implying: r d r x dt x. This is the Fundamental Lemma of Calculus of Variation (FLCV). Consider discounting and write the present-value Hamiltonian in the conventional CV form: H * (c,k,k,λ) = e -ρt {u(c) + λ[f(k)-k-c-k ]}= e -ρt H, where λ=λ * e ρt and H * / λ * =0 by construction Regarding r as H * and x as k, we can then apply FLCV to obtain: H k * d dt H k * d dt ( equation associated with k ( = λρ H/ k) * H ), or, ( ), which yields the Euler k