卢卡斯模型Lucas

Yi
=
Y
( Pi )−η P
exp(zi)
(7)
In log terms
yi = y + zi − η(pi − p), η > 0 (8)
Where y is the log aggregate real income,
zi is the shock to the demand for goods i, and η is the elasticity of demand across
)
1 γ−1
P
(5)
• Letting lower case letters denote the logarithms of the corresponding upper-case
variables, we can rewrite this condition as
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li = yi = γ − 1(pi − p)
Yi = Li
(1)
Where Li is the amount that the individual works and Yi the amount he or she produces.
• The budget constraint for the individual is given by
• So when the prices of the producer’s good increase, there is some chances that the increase reflects as rise in the price level, and some chance that it reflects a rise the goods’ relative price.
P Ci = PiYi
(2)
Where P is a price index of an aggregate
consumption good, which will be discussed
later, Ci is the individual i’s consumption of that aggregate consumption good, PiYi is the individual’s income or revenue.
• Individuals produce goods using their own labor, sell their output in competitive market, and use the proceeds to buy other producers’ output.
• Two shocks, preference shocks that change the relative demands for different goods, and monetary shocks. If the monetary shocks are observed, they change only the aggregate price level and have no real effect, but when they are not observed, we will show that they can change both the price level and aggregate output.
• That is, when a producer observes a change in the prices of his or her product, he or she does not knows it reflects a change
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in the goods’ relative price or a change in the aggregate price level. A change in the relative price alters the optimal amount to produce. A change in the aggregate price level, on the other hand, leaves optimal production unchanged.
• To account for the empirical evidence on the short run impact of money, we need either to introduce new channels through which money can affect real equilibrium under flexible price or consider nominal rigidities, or some kinds of barriers or limitations to the adjustment of nominal prices or wages.
each individual good. yi is the demand
for good i. The zis across goods, i.e.
∫h01azvidei
a mean of zero = 0. Also we
assumed that zi ∼ N (0, σz2). And the log
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Perfect information
• There are many different goods in the economy. Consider a representative producer of a typical good, goods i. The individual’s production function is simply
• Otherwise, a monetary change results only in proportional changes in all prices with no impact on real prices or quantities.
• Before discussing sticky price model, we investigate a flexible price model, which also can generate real effect of money shock in short run.
Basic framework
• The economy is populated by a large number of farmers, indexed by i ∈ (0, 1). Each farmer lives in a small island i and produces his own product.
• From Friedmand’s seminal book and VAR analysis, we know that money matters. Empirical evidence in US and other OECD countries shows that positive monetary shocks lead to a hump shaped positive response.
(6)
Thus, the individual’s labor supply and production are increasing in the relative price of his or her products.
Demand for individual goods
• The demand for a given good is assumed to depend on three factors: real income,the good’s relative price, and a random disturbance to preference. For tractability, demand is log-linear. So the demand for individual good i is given by:
• The nominal imperfection in Lucas model is that producers do not observe the aggregate price level; as a result, they make their production decisions without full knowledge of the relative prices that they will receive for their goods.
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• For simplicity, utility take the form
Ui
=
Ci
−
1 γ
Lγi
(3)
where γ > 1. Thus there is a constant marginal utility of consumption and increasing disutility of work. When the aggregate P is known, the individual’s maximization problem is simple. we can rewrite individual i’s utility as
• The rational response for the producer is to attribute part of the change to an increase in the price level and part to an increase in the relative prices, and therefore to increase output somewhat.
2
Flexible prices
• The first approach focuses on misperceptions about aggregate economics conditions; the second approach focuses on trading restrictions in financial market.
• The imperfect-inforLeabharlann Baiduation model is based on Lucas 1972 and Phelps 1970, which maintains the flexible price assumption. Friedman (1968, 1977) also have similar idea.
Ui
=
PiLi P
−
1 γ
Lγi
(4)
• As market is competitive, the individual
chooses Li to maximize utility, taking Pi and P as given. The first order condition
is
Li
=
(Pi
Econ 5140 Macroeconomic Analysis
Fall 2014 Jenny Xu HKUST Lucas Imperfect Information Model
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Introduction
• A major limitation of RBC model is their omission of any role for monetary changes in driving macroeconomic fluctuations.
of the aggregate output is given by
∫1
y = yidi = y¯i
(9)
0
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Thus, the aggregate price level is defined
as:
∫1
p = pidi = p¯i
0
(10)
Intuition for the demand structure is simple. The demand for a good is higher when total income is higher, when its prices is low relative to other prices, and when individuals have strong preferences for it.