The Complex Networks of Economic Interactions Essays in Agent-Based
Understanding the mechanism of a socio-economic system requires more than an understanding of the individuals that comprise the system. It also requires understanding how individuals interact with each other, and how the agg- gated outcome can be more tha
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$NLUD1DPDWDPH 7DLVHL.DL]RXML 0 say. This input is initially zero and graudally reach its full force after about the time elapse of 4/µ. Its Laplace transform is U (s) = µ/[s(s + µ)]. As a simple illustration of the difference of these two types of input signals on y(t), we assume that dynamics are described by a second order ordinary constant coefficient differential equation with zero initial conditions; y(0) = 0, and dy(0)/dt = 0. To be very concrete suppose that H(s) = (s + a)(s + b) with some positive a and b. This is the transfer function of a dynamic system
Why Macroeconomic Price Indices are Sluggish in Large Economies ?
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described by a second order differential equation with two stable eigenvalues −a, and −b. With the step input, the dynamic response is obtained by taking the inverse Laplace transform of Y (s) =
1 C A B = + + , s(s + a)(s + b) s s+a s+b
where A, B, and Care the constants, C = 1/ab, A = −1/[(b − a)a], and B = 1/[(b − a)b]. The time response of the pair of input and output with this transfer function is given by y(t) = C + Ae−at + Be−bt . This y(t) expression shows the multiplier effects of this block or unit of dynamics with the indicated transfer function. If a < b, then after the time span of about 4/a units of time, the output nearly settles to a constant, y(t) ≈ 1/ab.6 It takes about this much time for the effect of a sudden application of a step signal at the input to settle down at the output of the model. With the other input with a gradually rising magnitude such as u(t) = 1 − e−µt , with µ a positive constant much smaller than a and b, then y(t) is approximately equal to y(t) =
µ e−at e−bt µ . [− + ]+ b−a a−µ b−µ (a − µ)(b − µ)]e−µt
The first two exponential terms are due to the dynamic multiplier effects, and the third term is due to information transmission delay when u(t) gradually appear at the input terminal of this block or unit with the second-order dynamics. This expression is approximately equal to the last term above when µ is much smaller than a or b. The signal y(t) ≈ (1/ab)(1 − e−λt ), which takes a long time to reach its steady state value. In this case it is the behavior of input, not the dynamics, that causes the sluggish output. 3.2 Stochastic Spread of News in Trees: An example We next turn to the second type of lags that exist in trees with several levels of nodes. To illustrate our idea simply, we consider two simple economies with four sectors which are organized in two different ways. One is organized as a one level tree, and the other as a two level tree. Two-level trees are generally more sluggish in response than one-level trees. More gene
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