The correlation between hours worked and the real wage is close to zero.
Christiano and Eichenbaum 1992 "Current Real-Business-Cycle Theories and Aggregate Labor-Market Fluctuations"
http://ideas.repec.org/a/aea/aecrev/v82y1992i3p430-50.html
2012年3月29日木曜日
2012年3月28日水曜日
Recursive Competitive Equilibrium
"Formally, a Recursive Competitive Equilibrium (RCE) is characterized by time invariant functions of a limited number of ‘state variables’, which summarize the effects of past decisions and current information.
These functions (decision rules) include
(a) a pricing function,
(b) a value function,
(c) a period allocation policy specifying the individual’s decision,
(d) period allocation policy specifying the decision of each firm and
(e) a function specifying the law of motion of the capital stock."
Mehra 2005 "Recursive Competitive Equilibrium"
www.academicwebpages.com/preview/mehra/pdf/REC Nov 9.pdf
These functions (decision rules) include
(a) a pricing function,
(b) a value function,
(c) a period allocation policy specifying the individual’s decision,
(d) period allocation policy specifying the decision of each firm and
(e) a function specifying the law of motion of the capital stock."
Mehra 2005 "Recursive Competitive Equilibrium"
www.academicwebpages.com/preview/mehra/pdf/REC Nov 9.pdf
2012年3月23日金曜日
RBC: Impulse/ Amplification/ Propagation Mechanism
1. Impulse Mechanism
Exogenous variables in RBC models.
In the standard RBC models, the technology shock is only considered for impulse mechanisms. The standard RBC models generate the high correlation between productivity and labor; however, in the data this correlation is close to zero. In response to this problem, the introduction of government expenditure shocks, household production shocks, distortionary tax shock, etc. are considered.
Only technology shock means that in the labor market the labor demand is only affected by the shock, which generates the high correlation between productivity and labor. In order to weaken the correlation, some shocks which affect the labor supply is needed.
2. Amplification Mechanism
Typically the labor response but also could include variable capacity utilization of capital.
Hansen 1985 Indivisible Labor and the Business Cycle
"Unlike previous equilibrium models of the business cycle, this economy displays large fluctuations in hours worked and relatively small fluctuations in productivity."
http://www.sciencedirect.com/science/article/pii/030439328590039X
3. Propagation Mechanism
Typically the consumption and investment response to smooth the consumption path.
Cogley and Nason 1995 Output Dynamics in Real-Business-Cycle Models
"Many RBC models have weak internal propagation mechanisms and must rely
on external sources of dynamics to replicate both facts. Models that incorporate
labor adjustment costs are partially successful. They endogenously generate
positive autocorrelation in output growth, but they need implausibly large
transitory shocks to match the trend-reverting component in output."
"Robert E. Lucas and Thomas J. Sargent (1981) noted that capital accumulation and costs of adjustment could turn serially uncorrelated shocks into serially correlated movements in output. Although RBC theorists have explored this idea in great detail, the propagation mechanisms embodied in current models do not generate the right kind of output dynamics. Our results suggest that RBC theorists ought to devote further attention to modeling internal sources of propagation."
http://ideas.repec.org/a/aea/aecrev/v85y1995i3p492-511.html
Exogenous variables in RBC models.
In the standard RBC models, the technology shock is only considered for impulse mechanisms. The standard RBC models generate the high correlation between productivity and labor; however, in the data this correlation is close to zero. In response to this problem, the introduction of government expenditure shocks, household production shocks, distortionary tax shock, etc. are considered.
Only technology shock means that in the labor market the labor demand is only affected by the shock, which generates the high correlation between productivity and labor. In order to weaken the correlation, some shocks which affect the labor supply is needed.
2. Amplification Mechanism
Typically the labor response but also could include variable capacity utilization of capital.
Hansen 1985 Indivisible Labor and the Business Cycle
"Unlike previous equilibrium models of the business cycle, this economy displays large fluctuations in hours worked and relatively small fluctuations in productivity."
http://www.sciencedirect.com/science/article/pii/030439328590039X
3. Propagation Mechanism
Typically the consumption and investment response to smooth the consumption path.
Cogley and Nason 1995 Output Dynamics in Real-Business-Cycle Models
"Many RBC models have weak internal propagation mechanisms and must rely
on external sources of dynamics to replicate both facts. Models that incorporate
labor adjustment costs are partially successful. They endogenously generate
positive autocorrelation in output growth, but they need implausibly large
transitory shocks to match the trend-reverting component in output."
"Robert E. Lucas and Thomas J. Sargent (1981) noted that capital accumulation and costs of adjustment could turn serially uncorrelated shocks into serially correlated movements in output. Although RBC theorists have explored this idea in great detail, the propagation mechanisms embodied in current models do not generate the right kind of output dynamics. Our results suggest that RBC theorists ought to devote further attention to modeling internal sources of propagation."
http://ideas.repec.org/a/aea/aecrev/v85y1995i3p492-511.html
Equity Premium Puzzle/ Risk Free Rate Puzzle/ Lucas Cost of Business Cycles
1. Equity Premium Puzzle
With standard time separable and constant relative risk averse preferences, the consumption based asset pricing model is not consistent with the differences in average return between stocks (equity return) and bonds (risk free return).
Equity return >> Risk Free Return
To be data consistent, the relative risk aversion (RRA) parameter must be very high.
According to the US data: the return on the S&P500 from 1889 to 1978 and the yield on government bonds,
Equity Return ー Bond (Risk Free) Return 〜 6%
http://www.econ.yale.edu/smith/econ510a/book9.pdf
2. Risk Free Rate Puzzle
When RRA parameter is high, the risk free rate must be also high. This is not consistent with the real data.
<Mehra Prescott Model 1985>
qt=(1+r)^(-1)=βE[1/(λ^γ)]
λ: Consumption growth rate, γ: Relative risk aversion parameter
When γ is high, the right hand is low (λ>1), and r must be high.
3. Lucas Cost of Business Cycles
Based on Mehra Prescott model, he found that business cycle costs are very low because agents would pay a very small amount to insure their consumption against the business cycle volatility.
In the Mehra Prescott model, there must be some problem in preference specification. Hence, it is not surprising that Business cycle risk is quite small.
With standard time separable and constant relative risk averse preferences, the consumption based asset pricing model is not consistent with the differences in average return between stocks (equity return) and bonds (risk free return).
Equity return >> Risk Free Return
To be data consistent, the relative risk aversion (RRA) parameter must be very high.
According to the US data: the return on the S&P500 from 1889 to 1978 and the yield on government bonds,
Equity Return ー Bond (Risk Free) Return 〜 6%
http://www.econ.yale.edu/smith/econ510a/book9.pdf
2. Risk Free Rate Puzzle
When RRA parameter is high, the risk free rate must be also high. This is not consistent with the real data.
<Mehra Prescott Model 1985>
qt=(1+r)^(-1)=βE[1/(λ^γ)]
λ: Consumption growth rate, γ: Relative risk aversion parameter
When γ is high, the right hand is low (λ>1), and r must be high.
3. Lucas Cost of Business Cycles
Based on Mehra Prescott model, he found that business cycle costs are very low because agents would pay a very small amount to insure their consumption against the business cycle volatility.
In the Mehra Prescott model, there must be some problem in preference specification. Hence, it is not surprising that Business cycle risk is quite small.
Asset Price Basic Equations
1. Equity Price
q(t)=βE[u'(c(t+1))(q(t+1)+x(t+1))]/u'(c(t))
2. Risk Free Bond Price
p1(t)=βE[u'(c(t+1))]/u'(c(t))
3. Two-Period Bond Price
p2(t)=β^2E[u'(c(t+2))]/u'(c(t))
4. One-Period Forward Contract Price
pf(t)=βE[u'(c(t+2))]/E[u'(c(t+1))]
5. Equity Premium
EP(t)=E[q(t+1)+x(t+1)]/q(t)-1/p1(t)
6. Term Premium
TP(t)=E[p1(t+1)]/p2(t)-1/p1(t)
These equations are derived from the following DP:
State Variables: e(t-1) b1(t-1) b2(t-1) f(t-1) f(t-2) x(t)
Control Variables: c(t) e(t) b1(t) b2(t) f(t)
V(e(t-1) b1(t-1) b2(t-1) f(t-1) f(t-2) x(t))
=max{u(c(t))+βEV(e(t) b1(t) b2(t) f(t) f(t-1) x(t+1))
+λ(t)[(q(t)+x(t))e(t-1)+b1(t-1)+p1(t)b2(t-1)+f(t-2)
-c(t)-q(t)e(t)-p1(t)b1(t)-p2(t)b2(t)-pf(t-1)f(t-1)]}
If the endowment (x) is distributed as i.i.d,
(i) equity, risk free bond and two-period bond prices are positively correlated with the (current) endowment, and
(ii) the price of the one-period forward contract is constant,
on the ground that
(i) the numerators in the three prices are constant (i.i.d. expected values) and the denominators are increasing in the endowment, while
(ii) both the numerator and denominator of the one-period forward contract price are expected values.
Equity Premium
sign[EP]=-sign[cov(u'(c(t+1)),q(t+1)+x(t+1))]
Abel (1999) "Risk premia and term premia in general equilibrium"
In the standard Lucas fruit-tree model, the risk premium is necessarily positive.
http://www.sciencedirect.com/science/article/pii/S0304393298000397
Term Premium
sign[TP]=-sign[cov(u'(c(t+1)),p1(t+1))]
If the endowment (x) has positive serial correlation, then the sign of the term premium is indeterminate; in other words, the sign can be negative.
If the endowment (x) has negative or no serial correlation, then the sign of the term premium is positive.
http://www2.wiwi.hu-berlin.de/wpol/html/asset/pdf/lec02_LucasAssetPricing.pdf
q(t)=βE[u'(c(t+1))(q(t+1)+x(t+1))]/u'(c(t))
2. Risk Free Bond Price
p1(t)=βE[u'(c(t+1))]/u'(c(t))
3. Two-Period Bond Price
p2(t)=β^2E[u'(c(t+2))]/u'(c(t))
4. One-Period Forward Contract Price
pf(t)=βE[u'(c(t+2))]/E[u'(c(t+1))]
5. Equity Premium
EP(t)=E[q(t+1)+x(t+1)]/q(t)-1/p1(t)
6. Term Premium
TP(t)=E[p1(t+1)]/p2(t)-1/p1(t)
These equations are derived from the following DP:
State Variables: e(t-1) b1(t-1) b2(t-1) f(t-1) f(t-2) x(t)
Control Variables: c(t) e(t) b1(t) b2(t) f(t)
V(e(t-1) b1(t-1) b2(t-1) f(t-1) f(t-2) x(t))
=max{u(c(t))+βEV(e(t) b1(t) b2(t) f(t) f(t-1) x(t+1))
+λ(t)[(q(t)+x(t))e(t-1)+b1(t-1)+p1(t)b2(t-1)+f(t-2)
-c(t)-q(t)e(t)-p1(t)b1(t)-p2(t)b2(t)-pf(t-1)f(t-1)]}
If the endowment (x) is distributed as i.i.d,
(i) equity, risk free bond and two-period bond prices are positively correlated with the (current) endowment, and
(ii) the price of the one-period forward contract is constant,
on the ground that
(i) the numerators in the three prices are constant (i.i.d. expected values) and the denominators are increasing in the endowment, while
(ii) both the numerator and denominator of the one-period forward contract price are expected values.
Equity Premium
sign[EP]=-sign[cov(u'(c(t+1)),q(t+1)+x(t+1))]
Abel (1999) "Risk premia and term premia in general equilibrium"
In the standard Lucas fruit-tree model, the risk premium is necessarily positive.
http://www.sciencedirect.com/science/article/pii/S0304393298000397
Term Premium
sign[TP]=-sign[cov(u'(c(t+1)),p1(t+1))]
If the endowment (x) has positive serial correlation, then the sign of the term premium is indeterminate; in other words, the sign can be negative.
If the endowment (x) has negative or no serial correlation, then the sign of the term premium is positive.
http://www2.wiwi.hu-berlin.de/wpol/html/asset/pdf/lec02_LucasAssetPricing.pdf
2012年3月22日木曜日
Cyclical Properties of Real Data and RBC (Hansen Wright 1992)
Cyclical Properties of U.S. and Model-Generated Time Series (Hansen Wright 1992)
Type of Data or Model σy σc/σy σi/σy σh/σy σw/σy σh/σw cor(h,w)
U.S. Time Series*
Output 1.92 .45 2.78
Hours Worked:
1. Household Survey .78 .57 1.37 .07
(All Industries)
2. Establishment Survey .96 .45 2.15 -.14
(Nonag. Industries)
Models**
Standard 1.30 .31 3.15 .49 .53 .94 .93
Nonseparable Leisure 1.51 .29 3.23 .65 .40 1.63 .80
Indivisible Labor 1.73 .29 3.25 .76 .29 2.63 .76
Government Spending 1.24 .54 3.08 .55 .61 .90 .49
Home Production 1.71 .51 2.73 .75 .39 1.92 .49
·U.S. data here are the same as those in Table 2; they are 101 the longer time period: 1947:1-1991:3.
http://minneapolisfed.org/publications_papers/pub_display.cfm?id=242
Type of Data or Model σy σc/σy σi/σy σh/σy σw/σy σh/σw cor(h,w)
U.S. Time Series*
Output 1.92 .45 2.78
Hours Worked:
1. Household Survey .78 .57 1.37 .07
(All Industries)
2. Establishment Survey .96 .45 2.15 -.14
(Nonag. Industries)
Models**
Standard 1.30 .31 3.15 .49 .53 .94 .93
Nonseparable Leisure 1.51 .29 3.23 .65 .40 1.63 .80
Indivisible Labor 1.73 .29 3.25 .76 .29 2.63 .76
Government Spending 1.24 .54 3.08 .55 .61 .90 .49
Home Production 1.71 .51 2.73 .75 .39 1.92 .49
·U.S. data here are the same as those in Table 2; they are 101 the longer time period: 1947:1-1991:3.
http://minneapolisfed.org/publications_papers/pub_display.cfm?id=242
2012年3月21日水曜日
Brouwer Fixed Point Theorem
Every continuous function f from a convex compact subset K of a Euclidean space to K itself has a fixed point.
http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
Jensen's inequality
E[f(x)] ≥ f(E[x])
where f is convex function.
If f is concave,
E[f(x)] ≤ f(E[x]).
http://en.wikipedia.org/wiki/Jensen%27s_inequality
where f is convex function.
If f is concave,
E[f(x)] ≤ f(E[x]).
http://en.wikipedia.org/wiki/Jensen%27s_inequality
The Convergence Speed in Solow Growth Model
The speed of convergence in Solow growth model is
Δk/k=-λ[ln(k)-ln(k*)],
Δy/y=-λ[ln(y)-ln(y*)],
where λ≡(1-α)(n+g+δ).
"n+g+δ is typically about 6% per year (1-2% population growth, 1-2% growth in output per worker, and 3-4% depreciation). If capital's share is roughly one-third, λ is thus roughly 4%. k and y therefore move 4% of the remaining distance toward k* and y* each year, and take approximately 18 years to get halfway to their balanced-growth path values."
Romer, "Advanced Macroeconomics"
Δk/k=-λ[ln(k)-ln(k*)],
Δy/y=-λ[ln(y)-ln(y*)],
where λ≡(1-α)(n+g+δ).
"n+g+δ is typically about 6% per year (1-2% population growth, 1-2% growth in output per worker, and 3-4% depreciation). If capital's share is roughly one-third, λ is thus roughly 4%. k and y therefore move 4% of the remaining distance toward k* and y* each year, and take approximately 18 years to get halfway to their balanced-growth path values."
Romer, "Advanced Macroeconomics"
Saving Rate Differenece in Solow Growth Model
The elasticity of output with respect to the saving rate is α/(1-α)
(Check y*=(k*)^α=(s/[n+g+δ])^(α/[1-α])!)
Thus, " a 10 percent increase in the saving rate raises output per worker in the long run by about 5 percent relative to the path it would have followed."
Romer, "Advanced Macroeconomics"
(Check y*=(k*)^α=(s/[n+g+δ])^(α/[1-α])!)
Thus, " a 10 percent increase in the saving rate raises output per worker in the long run by about 5 percent relative to the path it would have followed."
Romer, "Advanced Macroeconomics"
Two Sources of Variation in Solow Growth Model
1. Capital per worker (K/L)
2. Effectiveness of Labor (A)
Romer, "Advanced Macroeconomics"
2. Effectiveness of Labor (A)
Romer, "Advanced Macroeconomics"
Problems in Solow Growth Model
"There are two problems with trying to account for large differences in incomes on the basis of differences in capital."
1. Required differences in capital are far too large.
10 times difference in output per worker → 1000 times difference in capital per worker! (α=1/3)
"Although capital-output rations vary somewhat across countries, the variation is not great. For example the capital-output ratio appears to be two to three times larger in the United States than in India."
2. Required differences in capital return are far too large.
10 times difference in output per worker → 100 times difference in capital return! (α=1/3, MPK=αy^[(α-1)/α])
"There is no evidence of such differences in rates of return. Direct measurement of returns on fincancial assets, for example, suggests only moderate variation over time and across countries."
Romer, "Advanced Macroeconomics"
1. Required differences in capital are far too large.
10 times difference in output per worker → 1000 times difference in capital per worker! (α=1/3)
"Although capital-output rations vary somewhat across countries, the variation is not great. For example the capital-output ratio appears to be two to three times larger in the United States than in India."
2. Required differences in capital return are far too large.
10 times difference in output per worker → 100 times difference in capital return! (α=1/3, MPK=αy^[(α-1)/α])
"There is no evidence of such differences in rates of return. Direct measurement of returns on fincancial assets, for example, suggests only moderate variation over time and across countries."
Romer, "Advanced Macroeconomics"
Bellman Principle of Optimality
An optimal policy has the property that, whatever the initial state and decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
2012年3月20日火曜日
Kronecker Products
1. (A ⊗ B)(C ⊗ D) = AC ⊗ BD.
2. (A ⊗ B)^(−1) = A^(−1) ⊗ B^(−1).
3. (A ⊗ B)′ = A′ ⊗ B′.
4. A ⊗ (B + C) = A ⊗ B + A ⊗ C.
2. (A ⊗ B)^(−1) = A^(−1) ⊗ B^(−1).
3. (A ⊗ B)′ = A′ ⊗ B′.
4. A ⊗ (B + C) = A ⊗ B + A ⊗ C.
2012年3月19日月曜日
Kaldor's Growth Facts
1. Output per worker (Y/L) and capital per worker (K/L) grow over time at relatively constant and positive rate.
2. The ratio between capital and output (K/Y) is relatively constant over time.
3. The real return to capital (r and r-δ) is relatively constant over time.
4. The capital share (rK/Y) and labor share (wL/Y) are roughly constant over time.
http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7768&rep=rep1&type=pdf
2. The ratio between capital and output (K/Y) is relatively constant over time.
3. The real return to capital (r and r-δ) is relatively constant over time.
4. The capital share (rK/Y) and labor share (wL/Y) are roughly constant over time.
http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7768&rep=rep1&type=pdf
Cartesian Product of Convex/ Compact
1. Any Cartesian product of convex sets is convex.
meboo.convexoptimization.com/BOOK/convexgeometry.pdf
2. Any Cartesian product of compact sets is compact.
www.ms.unimelb.edu.au/~rubin/math127/compact.pdf
meboo.convexoptimization.com/BOOK/convexgeometry.pdf
2. Any Cartesian product of compact sets is compact.
www.ms.unimelb.edu.au/~rubin/math127/compact.pdf
Minimization/ Maximization Necessary Conditions
1. Minimization Problem
If f(x) is quasiconvex, gi(x) is quasiconvex and hj(x) is affine (linear), then the necessary conditions are also sufficient conditions.
2. Maximization Problem
If f(x) is quasiconcave, gi(x) is quasiconvex and hj(x) is affine (linear), then the necessary conditions are also sufficient conditions.
objective function: f(x)
inequality constraints: gi(x)≦0
equality constraints: hj(x)=0
符号に注意!!
http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions
http://en.wikipedia.org/wiki/Convex_optimization
If f(x) is quasiconvex, gi(x) is quasiconvex and hj(x) is affine (linear), then the necessary conditions are also sufficient conditions.
2. Maximization Problem
If f(x) is quasiconcave, gi(x) is quasiconvex and hj(x) is affine (linear), then the necessary conditions are also sufficient conditions.
objective function: f(x)
inequality constraints: gi(x)≦0
equality constraints: hj(x)=0
符号に注意!!
http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions
http://en.wikipedia.org/wiki/Convex_optimization
Concave is Negative Convex
A function f(x) is concave over a convex set if and only if the function −f(x) is a convex function over the set.
http://en.wikipedia.org/wiki/Concave_function
http://en.wikipedia.org/wiki/Concave_function
Concave/ Quasi-Concave
1. A (strictly) concave function is (strictly) quasi-concave.
2. A (strictly) convex function is (strictly) quasi-convex.
3. A linear function is both concave and convex.
Source:
https://files.nyu.edu/caw1/public/UMath/Handouts/ums11h22convexsetsandfunctions.pdf
2. A (strictly) convex function is (strictly) quasi-convex.
3. A linear function is both concave and convex.
Source:
https://files.nyu.edu/caw1/public/UMath/Handouts/ums11h22convexsetsandfunctions.pdf
Properties of Convex Functions
1. Positive multiple of convex function is convex
2. Sum of convex functions is convex
...
Source:
www.ee.ucla.edu/ee236b/lectures/functions.pdf
2. Sum of convex functions is convex
...
Source:
www.ee.ucla.edu/ee236b/lectures/functions.pdf
2012年3月18日日曜日
Marginal Rate of Substitution（限界代替率）
The marginal rate of substitution of good x for good y is the amount of good y that the consumer must be given to compensate her for a one-unit mariginal reduction in her consumption of good x.
Mas-Colell, Whiston and Green "Microeconomic Theory" (p.54)
2012年3月17日土曜日
企業の訳語
一般的に企業、あるいは会社と言いたい場合はcompanyあるいはcorporationが無難。
company
一般的な企業をさす言葉。法人格の有無を問わない。
corporation
一般的な企業をさす言葉。法人格がある。比較的大きな企業。
firm
専門家集団の組織（accounting firm, law firm）をさすことが多い。
enterprise
抽象的概念で用いる。
ソース
http://eng.alc.co.jp/newsbiz/hinata/2006/04/companycorporat.html
company
一般的な企業をさす言葉。法人格の有無を問わない。
corporation
一般的な企業をさす言葉。法人格がある。比較的大きな企業。
firm
専門家集団の組織（accounting firm, law firm）をさすことが多い。
enterprise
抽象的概念で用いる。
ソース
http://eng.alc.co.jp/newsbiz/hinata/2006/04/companycorporat.html
2012年3月12日月曜日
Lucas Asset Pricing Model (Lucas "Tree" Model)
モデル設定
1. Large number of identical agents.
2. Only durable good is identical trees, one for each agent.
3. Each tree yields fruit (dividends) at the beginning of period.
4. Fruit is not durable, but the tree is durable.
5. Each agent starts life with one tree.
6. Fruit is governed by Markov process.
Ljungqvist and Sargent (2004) Recursive Macroeconomic Theory, pp.399-400
モデル設定
1. Large number of identical agents.
2. Only durable good is identical trees, one for each agent.
3. Each tree yields fruit (dividends) at the beginning of period.
4. Fruit is not durable, but the tree is durable.
5. Each agent starts life with one tree.
6. Fruit is governed by Markov process.
Ljungqvist and Sargent (2004) Recursive Macroeconomic Theory, pp.399-400
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