Topic 5: The Difference Equation
|
|
- Jason Wilkinson
- 5 years ago
- Views:
Transcription
1 Topic 5: The Difference Equation Yulei Luo Economics, HKU October 30, 2017 Luo, Y. (Economics, HKU) ME October 30, / 42
2 Discrete-time, Differences, and Difference Equations When time is taken to be a discrete variable, so that the variable t is allowed to take integer values only, the concept of the derivative will no longer appropriate (it involves infinitesimal changes, dt) and the change in variables must be described by so called differences ( t). Accordingly, the techniques of difference equations need to be developed. We may describe the pattern of change of y by the following difference equations: where y t+1 = y t+1 y t. y t+1 = 2 (1) or y t+1 = 0.1y t (2) Luo, Y. (Economics, HKU) ME October 30, / 42
3 Solving a FO Difference Equation Iterative Method. For the FO case, the difference equation describes the pattern of y between two consecutive periods only. Given an initial value y 0, a time path can be obtained by iteration. Consider y t+1 y t = 2 with y 0 = 15, y 1 = y y 2 = y = y (2) and in general, for any period t, y t = y 0 + t (2) = t. (3) Luo, Y. (Economics, HKU) ME October 30, / 42
4 . Consider y t+1 = 0.9y t with y 0. By iteration, y 1 = 0.9y 0, y 2 = (0.9) 2 y 0 y t = (0.9) t y 0 (4) Consider the following homogeneous difference equation ( n ) t my t+1 ny t = 0 = y t+1 = y0, (5) m which can be written as a more general form y t = Ab t. (6) Luo, Y. (Economics, HKU) ME October 30, / 42
5 General Method to Solve FO Difference Equation Suppose that we are solving the FO DE: y t+1 + ay t = c (7) The general solution is the sum of the two components: a particular solution y p (which is any solution of the above DE) and a complementary function y c. Let s first consider the CF. We try a solution of the form y t = Ab t, Ab t+1 + aab t = 0 = b + a = 0 or b = a, which means that the CF should be y c = A ( a) t (8) Consider now the particular solution. We try the simplest form y t = k, k + ak = c = k = c 1 + a = the PI is y p = c 1 + a (a = 1). (9) Luo, Y. (Economics, HKU) ME October 30, / 42
6 (Conti.) If it happens that a = 1, try another solution form y t = kt, k (t + 1) + akt = c c = k = t at = c. y p = ct (a = 1). (10) The general solution is then y t = A ( a) t + c 1 + a or y t = A ( a) t + ct (a = 1) (11) Using the initial condition y t = y 0 when t = 0, we can easily determine the definite solution: y 0 = A + c 1 + a = A = y 0 c (a = 1) or 1 + a y 0 = A + c 0 = A = y 0 (a = 1) Luo, Y. (Economics, HKU) ME October 30, / 42
7 The Dynamic Stability of Equilibrium In the discrete-time case, the dynamic stability depends on the Ab t term. The dynamic stability of equilibrium depends on whether or not the CF (Ab t ) will tend to zero as t. We can divide the range of b into seven distinct regions: see Figure { Nonoscillatory if b > 0 Oscillatory if b < 0 ; { Divergent if b > 1 Convergent if b < 1 The role of A. First, it can produce a scale effect without changing the basic configuration of the time path. Second, the sign of A can affect the shape of the path: a negative A can produce a mirror effect as well as a scale effect. Luo, Y. (Economics, HKU) ME October 30, / 42
8 The Cobweb Model A variant of the market model: it treats Q s as a function not of the current price but of the price of the preceding time period, that is, the supply function is lagged or delayed. Q s,t = S (P t 1 ) (12) When this function interacts with a demand function of the form Q d,t = D (P t ),interesting price dynamics will appear. Assuming linear supply and demand functions, and the market equilibrium implies Q s,t = Q d,t (13) Q d,t = α βp t (α, β > 0) (14) Q s,t = γ + δp t 1 (γ, δ > 0). (15) Luo, Y. (Economics, HKU) ME October 30, / 42
9 (Conti.) In equilibrium, the model can be reduced to the following FO DE βp t + δp t 1 = α + δ = P t+1 + δ β P t = α + δ β (16) Consequently, we have P t = ( P 0 α + γ ) ( δ ) t + α + γ β + δ β β + δ. (17) The particular integral P = α+γ β+δ is the intertemporal equilibrium price of the model. We can rewrite the price dynamics as follows P t = ( P 0 P ) ( δ β ) t + P. (18) Luo, Y. (Economics, HKU) ME October 30, / 42
10 (Conti.) P 0 P can have both the scale effect and the mirror effect on the price dynamics. Given our model specification (δ, β > 0), we can deduce an oscillatory time path because δ β < 0. That s why we call the model the Cobweb model. The model has three possibilities of oscillation patterns: Explosive if δ > β Uniform if δ = β Damped if δ < β See Figure Luo, Y. (Economics, HKU) ME October 30, / 42
11 Nonlinear Difference Equations-The Qualitative-Graphic Approach When nonlinearity occurs in the case of FO DE models, we can use the graphic approach (Phase diagram) to analyze the properties of the DE. Consider the following nonlinear DEs y t+1 + y 3 t = 5 or y t+1 + sin y t ln y t = 3 = y t+1 = f (y t ) when y t+1 and y t are plotted against each other, the resulting diagram is a phase diagram and the curve corresponding to f is a phase line. See Figure The first two phase lines, f 1 and f 2, are characterized by positive slopes f 1 (0, 1) and f 2 > 1 and the remaining two, f 3 and f 4, are negatively sloped f 3 ( 1, 0) and f 4 < 1 Luo, Y. (Economics, HKU) ME October 30, / 42
12 (Conti.) For the phase line f 1, the iterative process leads from y 0 to y in a steady path, without oscillation. For the phase line f 2 (whose slope is greater than 1), a divergent path appears. For phase lines, f 3 and f 4, the slopes are negative. The oscillatory time paths appear. Summary: The algebraic sign of the slope of the phase line determines whether there will be oscillation, and the absolution value of its slope governs the question of convergence. Luo, Y. (Economics, HKU) ME October 30, / 42
13 The Solow Model Solow (1956): What do simple neoclassical assumptions imply about growth? The model is the starting point for almost all analyses of growth. Key assumptions include: The production function F has three factors: capital K, labor L, and technology A: Y = F (K, AL) (K, L, A > 0), (19) where F K, F L > 0, and F KK, F LL < 0 (diminishing returns to capital and labor). AL means effective labor. F is assumed to be constant return to scale in K and L: ( ) K Y = ALF AL, 1 = ALf (k) or y = f (k), (20) where y = Y / (AL) and f (k) = F (k, 1). F also satisfy: lim K (K, AL) K 0 =, lim F L (K, AL) =, L 0 lim K (K, AL) K = 0, lim L (K, AL) = 0, L F (0, AL) = 0 for all A and L. Luo, Y. (Economics, HKU) ME October 30, / 42
14 Solving the Solow Model Assume that there is only a representative agent in the economy. The Solow model can be formulated as follows: K t+1 = (1 δ) K t + sf (K t, A t L t ), (21) A t+1 = (1 + g) A t, (22) L t+1 = (1 + n) L t, (23) given K 0, A 0, and L 0. Note that in the continuous-time version, we get a system of differential equations. Define k = K AL and y = Y AL, we have the following first-order nonlinear difference equation about capital per effective labor unit (k): k t+1 = (1 δ) k t + sf (k t ). (1 + g) (1 + n) Luo, Y. (Economics, HKU) ME October 30, / 42
15 Steady States There exist two steady states (SS): k = 0 and k = k > 0 that satisfies: (1 δ) k + sf (k) k = (1 + g) (1 + n). When f (k) = k α, [ ] s 1/(1 α) k =. (1 + g) (1 + n) (1 δ) Graphic Analysis [see the phase diagram.] Policy implications: The effects of s, α, and δ on the steady state stock of capital per effective labor unit. Alternatively, we can use the linearization method to approximate the original nonlinear difference equation around the intertemporal equilibrium (the steady state) and then solve the resulting linear difference equation. Luo, Y. (Economics, HKU) ME October 30, / 42
16 Linear Approximation How to linearize a difference equation (DE): x t+1 = f (x t ), given an initial condition, x 0? Typically, linearize the equation around a SS x satisfying x = f (x): x t+1 x + f (x) (x t x). (24) If we treat this approximation as exact, we have the following first-order DE: x t+1 = ax t + b, (25) where a = f (x) and b = (1 a) x. The solution to this DE is x t = ( 1 a t ) x + a t x 0. If a < 1, then a t 0 as t so that x t x. Hence, the original nonlinear DE is locally stable as it is approximated around the SS. For the Solow model, we have k t = ( 1 a t ) k + a t (1 α) (1 δ) k 0, where a = α + (0, 1). (1 + g) (1 + n) Luo, Y. (Economics, HKU) ME October 30, / 42
17 Speed of Convergence How does the initial level of capital per capita affect growth rates? Convergence: Poor countries grow faster than rich countries. Divergence: Rich countries grow faster than poor countries. The Solow model predicts that poor countries with low k will grow fast because of decreasing returns to capital: g t,t+1 = k [ ] t+1 k t s f (k t ) (1 δ) = + k t (1 + g) (1 + n) k t (1 + g) (1 + n) 1, where ( ) f (kt ) d /dk t = f (k t ) k t f (k t ) k t kt 2 < 0 because f (k t ) = kt α is concave (i.e., f (k t ) is decreasing). This convergence is only observed among U.S. states, Canadian provinces, European regions, etc, but not observed among the countries of the world. Luo, Y. (Economics, HKU) ME October 30, / 42
18 Summary k is called globally asymptotically stable as k t converges to it from any initial positive capital stock, k 0. What does this imply about the aggregate variables: K t A t L t Since is constant, K t and A t L t grow at the same rate (1 + g) (1 + n) 1 g + n. Since F (, ) is homogeneous of degree 1, Y t also grows at the same rate g + n. Per capita output Y t L t grows at rate g. The steady state for k t corresponds to a balanced growth path (a sequence in which all of the variables grow at a constant rate) for the original variables. Luo, Y. (Economics, HKU) ME October 30, / 42
19 Second Order Difference Equation A second-order difference equation involves the second difference of y : 2 y t+2 = ( y t+2 ) = (y t+2 y t+1 ) where is the first difference. = (y t+2 y t+1 ) (y t+1 y t ) = y t+2 2y t+1 + y t, (26) Luo, Y. (Economics, HKU) ME October 30, / 42
20 SO Linear DEs with Constant Coeffi cients and Constant Term A simple variety of SO equation takes the form y t+2 + a 1 y t+1 + a 2 y t = c (27) We first discuss particular solution. As usual, try the simplest solution form y t = k, which means that y p = k = c 1 + a 1 + a 2 (1 + a 1 + a 2 = 0) (28) In case a 1 + a 2 = 1, try another solution form y t = kt, which means that y p = kt = c a t (29) Luo, Y. (Economics, HKU) ME October 30, / 42
21 (Conti.) We next discuss the complementary function which is the solution of the reduced homogenous equation (c = 0). As in the FO DE case, try the following solution form y t = Ab t = (30) Ab t+2 + a 1 Ab t+1 + a 2 Ab t = 0 = (31) b 2 + a 1 b + a 2 = 0 (32) This quadratic characteristic equation have two roots: b 1, b 2 = a 1 ± a1 2 4a 2 2 and both should appear in the general solution of the reduced DE. There are three possibilities. (33) Luo, Y. (Economics, HKU) ME October 30, / 42
22 Case 1 (distinct real roots) When a 2 1 4a 2 > 0, the CF can be written as y c = A 1 b t 1 + A 2 b t 2. (34) Example: Consider which means that b 1 = 1, b 2 = 2, y t+2 + y t+1 2y t = 12, (35) y t = A 1 + A 2 ( 2) t + 4t (36) where A 1 and A 2 can be determined by two initial conditions y 0 = 4 and y 1 = 5 : 4 = A 1 + A 2 and 5 = A 1 + A 2 ( 2) + 4 = A 1 = 3 and A 2 = 1. Luo, Y. (Economics, HKU) ME October 30, / 42
23 Case 2 (repeated real roots) When a 2 1 4a 2 = 0, the CF can be written as y c = A 3 b t + A 4 tb t. (37) Example: Consider which means that b 1 = b 2 = 3, y t+2 + 6y t+1 + 9y t = 4, (38) y t = A 3 ( 3) t + A 4 t ( 3) t (39) where A 1 and A 2 can be determined by two initial conditions y 0 and y 1. Luo, Y. (Economics, HKU) ME October 30, / 42
24 Case 3 (complex roots) When a 2 1 4a 2 < 0, b 1, b 2 = h ± vi where h = a 1 a 2 2 and v = 1 +4a 2 2. The CF is y c = A 1 b t 1 + A 2 b t 2 = A 1 (h + vi) t + A 2 (h vi) t. (40) De Moivre theorem implies that (h ± vi) t = R t (cos θt ± i sin θt) where R = h 2 + v 2 = a 2, cos θ = h R, sin θ = v R (41) Luo, Y. (Economics, HKU) ME October 30, / 42
25 (Conti.) The CF can be rewritten as y c = A 1 R t (cos θt + i sin θt) + A 2 R t (cos θt i sin θt) (42) = R t [(A 1 + A 2 ) cos θt + (A 1 A 2 ) i sin θt] = R t (A 5 cos θt + A 6 i sin θt) (43) where R and θ can be determined once h and v become known. Example: Consider y t y t = 5,which means that h = 0, v = 1 2, R = y c = ( ) = 1 2 2, cos θ = 0, sin θ = 1, θ = π = (44) 2 ( ) 1 t ( A 5 cos π 2 2 t + A 6i sin π ) 2 t. (45) Luo, Y. (Economics, HKU) ME October 30, / 42
26 The Convergence of Time Path The convergence of time path y is determined by the two characteristic roots of the SO DE. In Case 1 if b 1 > 1 and b 2 > 1, then both components in the CF will be explosive and y c must be divergent. if b 1 < 1 and b 2 < 1, then both components in the CF will converge to 0 as t goes to infinity, as will y c also. if b 1 > 1 and b 2 < 1, then A 2 b t 2 tend to converge to 0, while A 1b t 1 tends to deviate further from 0 and will eventually render the path divergent. Call the root with higher absolute value the dominant root since this root sets the tone of the time path. A time path will converge iff the dominant root is less than 1 in absolute value. The nondominant root also affects the time path, at least in the beginning periods. Luo, Y. (Economics, HKU) ME October 30, / 42
27 In Case 2 (repeated roots), for the term A 4 tb t, if b > 1, the b t term will be explosive. and the multiplicative t term also serves to intensify the explosiveness as t increases. if b < 1, the b t term will be converge. and the multiplicative t will offset the convergence as t increases. It turns out the damping force b t of will eventually dominant the exploding force t. Hence, the basic requirement for convergence is still that the root be less than 1 in absolution value. Luo, Y. (Economics, HKU) ME October 30, / 42
28 In Case 3 (complex roots), The term A 5 cos θt + A 6 i sin θt produces a fluctuation pattern of a periodic nature. Since time is discrete, the resulting path displays a sort of stepped fluctuation. The term R t determines the convergence of y : determines whether the stepped fluctuation is to be intensified or mitigated as t increases. Hence, the basic requirement for convergence is still that the root be less than 1 in absolution value. The fluctuation can be gradually narrowed down iff R < 1 (Note that R is just the absolute value of the complex roots h ± vi). Luo, Y. (Economics, HKU) ME October 30, / 42
29 Difference Equations System So far our dynamic analysis has focused on a single difference equation. However, some economic models may include a system of simultaneous dynamic equations in which several variables need to be handled. Hence, the solution method to solve such dynamic system need to be introduced. The dynamic system with several dynamic equations and several variables can be equivalent with a single higher order equation with a single variable. Hence, the solution of a dynamic system would still include a set of PI and CF, and the dynamic stability of the system would still depend on the absolution values (for difference equation system) of the characteristic roots in the CF. Luo, Y. (Economics, HKU) ME October 30, / 42
30 The Transformation of a Higher-order Dynamic Equation In particular, a SO difference equation can be rewritten as two simultaneous FOC equations in two variables. Consider the following example: y t+2 + a 1 y t+1 + a 2 y t = c (46) If we introduce an artificial new variable x, defined as x t = y t+1, we can then express the original SO equation by the following two FO DE x t+1 + a 1 x t + a 2 y t = c (47) y t+1 x t = 0 (48) Luo, Y. (Economics, HKU) ME October 30, / 42
31 Solving Simultaneous Dynamic Equations Suppose that we are given x t+1 + 6x t + 9y t = 4 (49) y t+1 x t = 0 (50) To solve this two-de system, we still need to seek the PI and the CF, and sum them to obtain the desired time paths of the two variables x and y. We first solve for the PI. As usual, try the constant solution: y t+1 = y t = y and x t+1 = x t = x = x = y = 1 4. Luo, Y. (Economics, HKU) ME October 30, / 42
32 For the CF, try the following function forms x t = mb t and y t = nb t where m and n are arbitrary constants and b represents the characteristic root. Next, we need to find the values of m, n, and b that satisfy the reduced version. Substituting these guessed solutions into the above dynamic system and cancelling out the common term b t gives (b + 6) m + 9n = 0 (51) m + bn = 0 (52) which is a linear homogeneous-equation system in m and n. We can rule out the uninteresting trivial solution (m = n = 0) by requiring that b b = b2 + 6b + 9 = 0 (53) This characteristic equation have two roots b (= b 1 = b 2 ) = 3. Luo, Y. (Economics, HKU) ME October 30, / 42
33 (Conti.) Given each b i (i = 1, 2), the above homogeneous equation implies that there will have an infinite number of solutions for (m, n) For this repeated-root case, we have m i = k i n i (54) x t = m 1 ( 3) t + m 2 t ( 3) t, y t = n 1 ( 3) t + n 2 t ( 3) t which must satisfy y t+1 = x t : n 1 ( 3) t+1 + n 2 (t + 1) ( 3) t+1 = m 1 ( 3) t + m 2 t ( 3) t = Setting n 1 = A 3 and n 2 = A 4 gives m 1 = 3 (n 1 + n 2 ), m 2 = 3n 2 x c = 3A 3 ( 3) t 3A 4 (t + 1) ( 3) t (55) y c = A 3 ( 3) t + A 4 t ( 3) t (56) Note that both time paths have the same ( 3) t term, so they both explosive oscillation. Luo, Y. (Economics, HKU) ME October 30, / 42
34 Matrix Notation We can analyze the above dynamic system by using matrix. The above two-equation system can be written as [ ][ ] [ ][ ] [ ] 1 0 xt xt 4 + = 0 1 y t y t 0 }{{}}{{}}{{}}{{}}{{} I u K v d (57) Try a constant PI first, [ ] x (I + K ) = d = y [ x y ] = (I + K ) 1 d = [ 1/4 1/4 ]. Luo, Y. (Economics, HKU) ME October 30, / 42
35 (Conti.) Next, try the CF [ ] [ ] [ mb t+1 m u = nb t+1 = b t+1 m and v = n n [ ] m (bi + K ) = 0 n To avoid the trivial solution, we must have where A i are arbitrary constants. bi + K = 0 = b = 3 = m i = k i n i, where n i = A i, m i = k i A i ] b t = Luo, Y. (Economics, HKU) ME October 30, / 42
36 (Conti.) With distinct real roots, [ ] [ xc m1 b = 1 t + m 2b t ] [ 2 k1 A y c n 1 b1 t + n 2b2 t = 1 b1 t + k 2A 2 b2 t A 1 b1 t + A 2b2 t ]. (58) With repeated roots, [ xc y c ] [ m1 b = 1 t + m 2tb2 t n 1 b1 t + n 2tb2 t ] (59) The general solution can be written as [ ] [ xt xc = y t y c ] + [ x y ]. (60) Luo, Y. (Economics, HKU) ME October 30, / 42
37 Two-Variable Phase Diagram: Discrete-time Case Now we shall discuss the qualitative-graphic (phase-diagram) analysis of a nonlinear difference equation system. Specifically, we focus on the following two-equation system x t+1 x t = f (x t, y t ) (61) y t+1 y t = g (x t, y t ) (62) which is called autonomous system (t is not an explicit argument in f and g). The two-variable phase diagram (PD) can answer the qualitative questions: the location and the dynamic stability of the intertemporal equilibrium. The most crucial task of the PD is to determine the direction of movement of the two variables over time. In the two-variable case, we can also draw the PD in the space of (x, y). Luo, Y. (Economics, HKU) ME October 30, / 42
38 (Conti.) In this case, we have two demarcation lines: x t+1 = x t+1 x t = f (x t, y t ) = 0 (63) y t+1 = y t+1 y t = g (x t, y t ) = 0 (64) which interact at point E representing the intertemporal equilibrium ( x t+1 = 0 and y t+1 = 0) and divide the space into 4 regions. (will be specified later.) If the demarcation line can be solved for y in terms of x, we can plot the line in the (x, y) space. Otherwise, we can use the implicit-function theorem to derive: slope of x t+1 = dy dx f / x x t+1 =0 = f / y = f x ; (65) f y slope of y t+1 = dy dx y t+1 =0 = g/ x g/ y = g x g y. (66) Specifically, we assume that f x < 0, f y > 0, g x > 0, g y < 0,which means that both slopes are positive. Further assume that f x fy > g x g y. Luo, Y. (Economics, HKU) ME October 30, / 42
39 (Conti.) The two curves, at any other point, either x or y changes over time according to the signs of x t+1 and y t+1 at that point: d ( x t+1 ) dx = f x < 0, (67) which means that as we move from west to east in the space (as x increases), x t+1 decrease so that the sign of x t+1 must pass through three stages, in the order: +, 0,. Similarly, d ( y t+1 ) dy = g y < 0, (68) which means that as we move from south to north in the space (as y increases), y t+1 decreases so that the sign of y t+1 must pass through three stages, in the order: +, 0,. Luo, Y. (Economics, HKU) ME October 30, / 42
40 Linearization of a Nonlinearization Difference-Equation System Another qualitative technique of analyzing a nonlinear difference equation system is to examine its linear approximation which is derived by using the Taylor expansion of the system around its intertemporal equilibrium. At the point of expansion (i.e., the IE), the linear approximation has the same equilibrium as the original nonlinear system. In a suffi ciently small neighborhood of E, the linear approximation should have the same general streamline configuration as the original system. As long as we confine our stability analysis to the immediate neighborhood of the IE, the approximated system can include enough information from the original nonlinear system. This analysis is called local stability analysis. Luo, Y. (Economics, HKU) ME October 30, / 42
41 (Conti.) For the two difference equation system, we have x t+1 = f (x 0, y 0 ) + f x (x 0, y 0 ) (x x 0 ) + f y (x 0, y 0 ) (y y 0 ) y t+1 = g (x 0, y 0 ) + g x (x 0, y 0 ) (x x 0 ) + g y (x 0, y 0 ) (y y 0 ) For purpose of local stability analysis, the above linearization can be put a simpler form. First, the expansion point is the IE, (x, y) and f (x, y) = g (x, y) = 0. We then have another form of linearization x t+1 (1 + f x (x, y)) x f y (x, y) y = f x (x, y) x f y (x, y) y y t+1 g x (x, y) x (1 + g y (x, y)) y = g x (x, y) x g y (x, y) y which means that [ xt+1 x y t+1 y ] [ ] 1 + fx f y g x 1 + g y (x,y ) }{{} J E [ xt x y t y ] = [ 0 0 ]. Luo, Y. (Economics, HKU) ME October 30, / 42
42 (Conti.) The Jacobian matrix JE in the above reduced system can determine the local stability of the equilibrium. Denote [ ] [ ] 1 + fx f J E = y a b = (69) g x 1 + g y c d (x,y ) The characteristic roots of the reduced linearization is r a b c r d = r 2 (a + d) r + (ad bc) = 0 = tr (J E ) = r 1 + r 2 = a + d = 2 + f x + g y (70) det (J E ) = r 1 r 2 = ad bc = (1 + f x ) (1 + g y ) f y g x = (71) r 1, r 2 = tr (J E ) ± (tr (J E )) 2 4 det (J E ) 2 There are also four cases for the local stability of the above system, but here we only focus on the most popular economic case: The saddle-point case in which r 1 > 1 and r 2 < 1. We will discuss the application in the next lecture on dynamic optimization. Luo, Y. (Economics, HKU) ME October 30, / 42
Lecture 6: Discrete-Time Dynamic Optimization
Lecture 6: Discrete-Time Dynamic Optimization Yulei Luo Economics, HKU November 13, 2017 Luo, Y. (Economics, HKU) ECON0703: ME November 13, 2017 1 / 43 The Nature of Optimal Control In static optimization,
More informationTopic 8: Optimal Investment
Topic 8: Optimal Investment Yulei Luo SEF of HKU November 22, 2013 Luo, Y. SEF of HKU) Macro Theory November 22, 2013 1 / 22 Demand for Investment The importance of investment. First, the combination of
More informationEC744 Lecture Notes: Economic Dynamics. Prof. Jianjun Miao
EC744 Lecture Notes: Economic Dynamics Prof. Jianjun Miao 1 Deterministic Dynamic System State vector x t 2 R n State transition function x t = g x 0 ; t; ; x 0 = x 0 ; parameter 2 R p A parametrized dynamic
More informationSuggested Solutions to Problem Set 2
Macroeconomic Theory, Fall 03 SEF, HKU Instructor: Dr. Yulei Luo October 03 Suggested Solutions to Problem Set. 0 points] Consider the following Ramsey-Cass-Koopmans model with fiscal policy. First, we
More informationDYNAMIC LECTURE 1 UNIVERSITY OF MARYLAND: ECON 600
DYNAMIC LECTURE 1 UNIVERSITY OF MARYLAND: ECON 6 1. differential Equations 1 1.1. Basic Concepts for Univariate Equations. We use differential equations to model situations which treat time as a continuous
More informationDynamical Systems. August 13, 2013
Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.
More informationAnalysis of the speed of convergence
Analysis of the speed of convergence Lionel Artige HEC Université de Liège 30 january 2010 Neoclassical Production Function We will assume a production function of the Cobb-Douglas form: F[K(t), L(t),
More informationEndogenous Growth Theory
Endogenous Growth Theory Lecture Notes for the winter term 2010/2011 Ingrid Ott Tim Deeken October 21st, 2010 CHAIR IN ECONOMIC POLICY KIT University of the State of Baden-Wuerttemberg and National Laboratory
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
More informationSolow Growth Model. Michael Bar. February 28, Introduction Some facts about modern growth Questions... 4
Solow Growth Model Michael Bar February 28, 208 Contents Introduction 2. Some facts about modern growth........................ 3.2 Questions..................................... 4 2 The Solow Model 5
More information2 Discrete Dynamical Systems (DDS)
2 Discrete Dynamical Systems (DDS) 2.1 Basics A Discrete Dynamical System (DDS) models the change (or dynamics) of single or multiple populations or quantities in which the change occurs deterministically
More informationAdvanced Macroeconomics
Advanced Macroeconomics The Ramsey Model Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 30 Introduction Authors: Frank Ramsey (1928), David Cass (1965) and Tjalling
More informationRamsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path
Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ryoji Ohdoi Dept. of Industrial Engineering and Economics, Tokyo Tech This lecture note is mainly based on Ch. 8 of Acemoglu
More informationHOMEWORK #1 This homework assignment is due at 5PM on Friday, November 3 in Marnix Amand s mailbox.
Econ 50a (second half) Yale University Fall 2006 Prof. Tony Smith HOMEWORK # This homework assignment is due at 5PM on Friday, November 3 in Marnix Amand s mailbox.. Consider a growth model with capital
More informationMath 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationEconomic Growth: Lecture 9, Neoclassical Endogenous Growth
14.452 Economic Growth: Lecture 9, Neoclassical Endogenous Growth Daron Acemoglu MIT November 28, 2017. Daron Acemoglu (MIT) Economic Growth Lecture 9 November 28, 2017. 1 / 41 First-Generation Models
More informationLecture notes on modern growth theory
Lecture notes on modern growth theory Part 2 Mario Tirelli Very preliminary material Not to be circulated without the permission of the author October 25, 2017 Contents 1. Introduction 1 2. Optimal economic
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo FBE, HKU September 2, 2018 Luo, Y. (FBE, HKU) ME September 2, 2018 1 / 35 Course Outline Economics: The study of the choices people (consumers, firm managers,
More informationECON0702: Mathematical Methods in Economics
ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 14, 2009 Luo, Y. (SEF of HKU) MME January 14, 2009 1 / 44 Comparative Statics and The Concept of Derivative Comparative Statics
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationChapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12
Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider
More informationAdvanced Macroeconomics
Advanced Macroeconomics The Ramsey Model Micha l Brzoza-Brzezina/Marcin Kolasa Warsaw School of Economics Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 47 Introduction Authors:
More informationSolution Methods. Jesús Fernández-Villaverde. University of Pennsylvania. March 16, 2016
Solution Methods Jesús Fernández-Villaverde University of Pennsylvania March 16, 2016 Jesús Fernández-Villaverde (PENN) Solution Methods March 16, 2016 1 / 36 Functional equations A large class of problems
More informationAssumption 5. The technology is represented by a production function, F : R 3 + R +, F (K t, N t, A t )
6. Economic growth Let us recall the main facts on growth examined in the first chapter and add some additional ones. (1) Real output (per-worker) roughly grows at a constant rate (i.e. labor productivity
More informationLecture 2 The Centralized Economy
Lecture 2 The Centralized Economy Economics 5118 Macroeconomic Theory Kam Yu Winter 2013 Outline 1 Introduction 2 The Basic DGE Closed Economy 3 Golden Rule Solution 4 Optimal Solution The Euler Equation
More informationEconomic Growth: Lecture 8, Overlapping Generations
14.452 Economic Growth: Lecture 8, Overlapping Generations Daron Acemoglu MIT November 20, 2018 Daron Acemoglu (MIT) Economic Growth Lecture 8 November 20, 2018 1 / 46 Growth with Overlapping Generations
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationComparative Statics. Autumn 2018
Comparative Statics Autumn 2018 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationProblem Set #2: Overlapping Generations Models Suggested Solutions - Q2 revised
University of Warwick EC9A Advanced Macroeconomic Analysis Problem Set #: Overlapping Generations Models Suggested Solutions - Q revised Jorge F. Chavez December 6, 0 Question Consider the following production
More informationLecture 5: The neoclassical growth model
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 3005 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 5: The neoclassical
More informationHonors Calculus Quiz 9 Solutions 12/2/5
Honors Calculus Quiz Solutions //5 Question Find the centroid of the region R bounded by the curves 0y y + x and y 0y + 50 x Also determine the volumes of revolution of the region R about the coordinate
More informationLecture 3 - Solow Model
Lecture 3 - Solow Model EC308 Advanced Macroeconomics 16/02/2016 (EC308) Lecture 3 - Solow Model 16/02/2016 1 / 26 Introduction Solow Model Sometimes known as Solow-Swan Model: Solow (1956): General Production
More informationMIDTERM 1 PRACTICE PROBLEM SOLUTIONS
MIDTERM 1 PRACTICE PROBLEM SOLUTIONS Problem 1. Give an example of: (a) an ODE of the form y (t) = f(y) such that all solutions with y(0) > 0 satisfy y(t) = +. lim t + (b) an ODE of the form y (t) = f(y)
More informationThe Growth Model in Continuous Time (Ramsey Model)
The Growth Model in Continuous Time (Ramsey Model) Prof. Lutz Hendricks Econ720 September 27, 2017 1 / 32 The Growth Model in Continuous Time We add optimizing households to the Solow model. We first study
More informationEconomic Growth
MIT OpenCourseWare http://ocw.mit.edu 14.452 Economic Growth Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 14.452 Economic Growth: Lecture
More informationEconomic Growth (Continued) The Ramsey-Cass-Koopmans Model. 1 Literature. Ramsey (1928) Cass (1965) and Koopmans (1965) 2 Households (Preferences)
III C Economic Growth (Continued) The Ramsey-Cass-Koopmans Model 1 Literature Ramsey (1928) Cass (1965) and Koopmans (1965) 2 Households (Preferences) Population growth: L(0) = 1, L(t) = e nt (n > 0 is
More informationFrom Difference to Differential Equations I
From Difference to Differential Equations I Start with a simple difference equation x (t + 1) x (t) = g(x (t)). (30) Now consider the following approximation for any t [0, 1], x (t + t) x (t) t g(x (t)),
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationB. Differential Equations A differential equation is an equation of the form
B Differential Equations A differential equation is an equation of the form ( n) F[ t; x(, xʹ (, x ʹ ʹ (, x ( ; α] = 0 dx d x ( n) d x where x ʹ ( =, x ʹ ʹ ( =,, x ( = n A differential equation describes
More informationAircraft Dynamics First order and Second order system
Aircraft Dynamics First order and Second order system Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Aircraft dynamic
More information1. Using the model and notations covered in class, the expected returns are:
Econ 510a second half Yale University Fall 2006 Prof. Tony Smith HOMEWORK #5 This homework assignment is due at 5PM on Friday, December 8 in Marnix Amand s mailbox. Solution 1. a In the Mehra-Prescott
More informationSmall Open Economy RBC Model Uribe, Chapter 4
Small Open Economy RBC Model Uribe, Chapter 4 1 Basic Model 1.1 Uzawa Utility E 0 t=0 θ t U (c t, h t ) θ 0 = 1 θ t+1 = β (c t, h t ) θ t ; β c < 0; β h > 0. Time-varying discount factor With a constant
More informationNew Notes on the Solow Growth Model
New Notes on the Solow Growth Model Roberto Chang September 2009 1 The Model The firstingredientofadynamicmodelisthedescriptionofthetimehorizon. In the original Solow model, time is continuous and the
More informationECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2
ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2 The due date for this assignment is Tuesday, October 2. ( Total points = 50). (Two-sector growth model) Consider the
More informationLecture 5 Dynamics of the Growth Model. Noah Williams
Lecture 5 Dynamics of the Growth Model Noah Williams University of Wisconsin - Madison Economics 702/312 Spring 2016 An Example Now work out a parametric example, using standard functional forms. Cobb-Douglas
More informationTime-varying Consumption Tax, Productive Government Spending, and Aggregate Instability.
Time-varying Consumption Tax, Productive Government Spending, and Aggregate Instability. Literature Schmitt-Grohe and Uribe (JPE 1997): Ramsey model with endogenous labor income tax + balanced budget (fiscal)
More informationEndogenous Growth: AK Model
Endogenous Growth: AK Model Prof. Lutz Hendricks Econ720 October 24, 2017 1 / 35 Endogenous Growth Why do countries grow? A question with large welfare consequences. We need models where growth is endogenous.
More informationMacroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models
Macroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models Prof. George McCandless UCEMA Spring 2008 1 Class 1: introduction and rst models What we will do today 1. Organization of course
More information14.05: Section Handout #1 Solow Model
14.05: Section Handout #1 Solow Model TA: Jose Tessada September 16, 2005 Today we will review the basic elements of the Solow model. Be prepared to ask any questions you may have about the derivation
More informationFunctions. A function is a rule that gives exactly one output number to each input number.
Functions A function is a rule that gives exactly one output number to each input number. Why it is important to us? The set of all input numbers to which the rule applies is called the domain of the function.
More informationA Note on the Ramsey Growth Model with the von Bertalanffy Population Law
Applied Mathematical Sciences, Vol 4, 2010, no 65, 3233-3238 A Note on the Ramsey Growth Model with the von Bertalanffy Population aw uca Guerrini Department of Mathematics for Economic and Social Sciences
More informationEquilibrium in a Production Economy
Equilibrium in a Production Economy Prof. Eric Sims University of Notre Dame Fall 2012 Sims (ND) Equilibrium in a Production Economy Fall 2012 1 / 23 Production Economy Last time: studied equilibrium in
More informationMathematical Economics: Lecture 9
Mathematical Economics: Lecture 9 Yu Ren WISE, Xiamen University October 17, 2011 Outline 1 Chapter 14: Calculus of Several Variables New Section Chapter 14: Calculus of Several Variables Partial Derivatives
More informationToulouse School of Economics, M2 Macroeconomics 1 Professor Franck Portier. Exam Solution
Toulouse School of Economics, 2013-2014 M2 Macroeconomics 1 Professor Franck Portier Exam Solution This is a 3 hours exam. Class slides and any handwritten material are allowed. You must write legibly.
More informationMath 215/255 Final Exam (Dec 2005)
Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationOn the Dynamic Implications of the Cobb- Douglas Production Function
From the SelectedWorks of Jürgen Antony 2010 On the Dynamic Implications of the Cobb- Douglas Production Function Jürgen Antony, CPB Netherlands Bureau for Economic Policy Analysis Available at: https://works.bepress.com/antony/7/
More informationIntroduction to Real Business Cycles: The Solow Model and Dynamic Optimization
Introduction to Real Business Cycles: The Solow Model and Dynamic Optimization Vivaldo Mendes a ISCTE IUL Department of Economics 24 September 2017 (Vivaldo M. Mendes ) Macroeconomics (M8674) 24 September
More informationLecture 1: The Classical Optimal Growth Model
Lecture 1: The Classical Optimal Growth Model This lecture introduces the classical optimal economic growth problem. Solving the problem will require a dynamic optimisation technique: a simple calculus
More informationChapter 4: First-order differential equations. Similarity and Transport Phenomena in Fluid Dynamics Christophe Ancey
Chapter 4: First-order differential equations Similarity and Transport Phenomena in Fluid Dynamics Christophe Ancey Chapter 4: First-order differential equations Phase portrait Singular point Separatrix
More informationEcon 204A: Section 3
Econ 204A: Section 3 Ryan Sherrard University of California, Santa Barbara 18 October 2016 Sherrard (UCSB) Section 3 18 October 2016 1 / 19 Notes on Problem Set 2 Total Derivative Review sf (k ) = (δ +
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More informationMacroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PS 5, preliminary version
Macroeconomics I, UPF Professor ntonio Ciccone SOUTIONS PS 5, preliminary version 1 The Solow K model with transitional dynamics Consider the following Solow economy: production is determined by Y F (K,
More information4- Current Method of Explaining Business Cycles: DSGE Models. Basic Economic Models
4- Current Method of Explaining Business Cycles: DSGE Models Basic Economic Models In Economics, we use theoretical models to explain the economic processes in the real world. These models de ne a relation
More informationSimple Iteration, cont d
Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 2 Notes These notes correspond to Section 1.2 in the text. Simple Iteration, cont d In general, nonlinear equations cannot be solved in a finite sequence
More informationA t = B A F (φ A t K t, N A t X t ) S t = B S F (φ S t K t, N S t X t ) M t + δk + K = B M F (φ M t K t, N M t X t )
Notes on Kongsamut et al. (2001) The goal of this model is to be consistent with the Kaldor facts (constancy of growth rates, capital shares, capital-output ratios) and the Kuznets facts (employment in
More informationEconomic Growth: Lectures 5-7, Neoclassical Growth
14.452 Economic Growth: Lectures 5-7, Neoclassical Growth Daron Acemoglu MIT November 7, 9 and 14, 2017. Daron Acemoglu (MIT) Economic Growth Lectures 5-7 November 7, 9 and 14, 2017. 1 / 83 Introduction
More informationPerturbation and Projection Methods for Solving DSGE Models
Perturbation and Projection Methods for Solving DSGE Models Lawrence J. Christiano Discussion of projections taken from Christiano Fisher, Algorithms for Solving Dynamic Models with Occasionally Binding
More informationChapter 9 Solow. O. Afonso, P. B. Vasconcelos. Computational Economics: a concise introduction
Chapter 9 Solow O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 27 Overview 1 Introduction 2 Economic model 3 Computational
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More informationMath 216 Final Exam 24 April, 2017
Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationAY Term 1 Examination November 2013 ECON205 INTERMEDIATE MATHEMATICS FOR ECONOMICS
AY203-4 Term Examination November 203 ECON205 INTERMEDIATE MATHEMATICS FOR ECONOMICS INSTRUCTIONS TO CANDIDATES The time allowed for this examination paper is TWO hours 2 This examination paper contains
More informationThe construction and use of a phase diagram to investigate the properties of a dynamic system
File: Phase.doc/pdf The construction and use of a phase diagram to investigate the properties of a dynamic system 1. Introduction Many economic or biological-economic models can be represented as dynamic
More informationFundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad
Fundamentals of Dynamical Systems / Discrete-Time Models Dr. Dylan McNamara people.uncw.edu/ mcnamarad Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian
More informationEconomic Growth: Lecture 7, Overlapping Generations
14.452 Economic Growth: Lecture 7, Overlapping Generations Daron Acemoglu MIT November 17, 2009. Daron Acemoglu (MIT) Economic Growth Lecture 7 November 17, 2009. 1 / 54 Growth with Overlapping Generations
More informationAdvanced Mathematics for Economics, course Juan Pablo Rincón Zapatero
Advanced Mathematics for Economics, course 2013-2014 Juan Pablo Rincón Zapatero Contents 1. Review of Matrices and Determinants 2 1.1. Square matrices 2 1.2. Determinants 3 2. Diagonalization of matrices
More informationModeling Economic Growth Using Differential Equations
Modeling Economic Growth Using Differential Equations Chad Tanioka Occidental College February 25, 2016 Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, 2016 1 / 28 Overview
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationDynamic (Stochastic) General Equilibrium and Growth
Dynamic (Stochastic) General Equilibrium and Growth Martin Ellison Nuffi eld College Michaelmas Term 2018 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term 2018 1 / 43 Macroeconomics is Dynamic
More informationMath 163: Lecture notes
Math 63: Lecture notes Professor Monika Nitsche March 2, 2 Special functions that are inverses of known functions. Inverse functions (Day ) Go over: early exam, hw, quizzes, grading scheme, attendance
More informationIntermediate Macroeconomics, EC2201. L2: Economic growth II
Intermediate Macroeconomics, EC2201 L2: Economic growth II Anna Seim Department of Economics, Stockholm University Spring 2017 1 / 64 Contents and literature The Solow model. Human capital. The Romer model.
More informationNumerical differentiation
Numerical differentiation Paul Seidel 1801 Lecture Notes Fall 011 Suppose that we have a function f(x) which is not given by a formula but as a result of some measurement or simulation (computer experiment)
More informationNeoclassical Models of Endogenous Growth
Neoclassical Models of Endogenous Growth October 2007 () Endogenous Growth October 2007 1 / 20 Motivation What are the determinants of long run growth? Growth in the "e ectiveness of labour" should depend
More informationFirst Order Linear Ordinary Differential Equations
First Order Linear Ordinary Differential Equations The most general first order linear ODE is an equation of the form p t dy dt q t y t f t. 1 Herepqarecalledcoefficients f is referred to as the forcing
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential
More informationStability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games
Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,
More informationThe Ramsey Model. Alessandra Pelloni. October TEI Lecture. Alessandra Pelloni (TEI Lecture) Economic Growth October / 61
The Ramsey Model Alessandra Pelloni TEI Lecture October 2015 Alessandra Pelloni (TEI Lecture) Economic Growth October 2015 1 / 61 Introduction Introduction Introduction Ramsey-Cass-Koopmans model: di ers
More informationADVANCED MACROECONOMICS I
Name: Students ID: ADVANCED MACROECONOMICS I I. Short Questions (21/2 points each) Mark the following statements as True (T) or False (F) and give a brief explanation of your answer in each case. 1. 2.
More informationErgodicity and Non-Ergodicity in Economics
Abstract An stochastic system is called ergodic if it tends in probability to a limiting form that is independent of the initial conditions. Breakdown of ergodicity gives rise to path dependence. We illustrate
More informationLecture 8: Aggregate demand and supply dynamics, closed economy case.
Lecture 8: Aggregate demand and supply dynamics, closed economy case. Ragnar Nymoen Department of Economics, University of Oslo October 20, 2008 1 Ch 17, 19 and 20 in IAM Since our primary concern is to
More informationMathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics
Mathematics for Economics ECON 53035 MA/MSSc in Economics-2017/2018 Dr. W. M. Semasinghe Senior Lecturer Department of Economics MATHEMATICS AND STATISTICS LERNING OUTCOMES: By the end of this course unit
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationChapter 4 Differentiation
Chapter 4 Differentiation 08 Section 4. The derivative of a function Practice Problems (a) (b) (c) 3 8 3 ( ) 4 3 5 4 ( ) 5 3 3 0 0 49 ( ) 50 Using a calculator, the values of the cube function, correct
More informationAN AK TIME TO BUILD GROWTH MODEL
International Journal of Pure and Applied Mathematics Volume 78 No. 7 2012, 1005-1009 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu PA ijpam.eu AN AK TIME TO BUILD GROWTH MODEL Luca Guerrini
More informationNonlinear Autonomous Systems of Differential
Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More information