1. Zhongbo Lian Class 979 Dec 16th, 2009
Q-Theory
Zhongbo Lian
Class 979
Dec 16th, 2009
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2. Summers’q-theory key assumptions :
1. The wage is such that the labor market is clearing.
2. There is a rising cost of new capital as investment becomes faster relative to the capital
stock.
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3. Introduction to Summers
　　　 萨默斯16岁时在MIT攻读物理。1982年，他获得哈佛大学经济学博士学位。次年，28岁的萨默斯成为哈佛历史上最年轻的终身教授。
　　 年，萨默斯成为里根政府的白宫经济顾问委员会成员。1991年，他出任世界银行副行长兼首席经济师。1993年，他进入财政部，担任负责国际事务的副部长。1999年7月，出任克林顿政府财政部长。2001年7月—2006年6月，萨默斯出任哈佛大学校长。现为白宫国家经济委员会主任。
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4. Conclusions
There are two ways that one can acquire a new unit of investment.
One way is to build a new unit of capital. The other way is to buy shares on the stock market which represent ownership of that capital.
What is the equilibrium:
the equilibrium level of investment according to q-theory then occurs when there are no arbitrage opportunities left.That occurs when the value of the shares of a firm per unit of its capital stock will exactly equal the marginal cost of a unit of investment.
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5. the value of the stock per unit of capital: V/K.
The cost of building a new unit of capital is the basic unit cost of the investment plus the marginal adjustment cost from constructing one more investment unit.
Tobin gave the ratio V/K the name q. In this theory investment will be determined by q.
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6. Microeconomic analysis
Assume that stock prices are determined by fundamentals.
V is then the PDV of the profits.The present discounted value of the returns from a unit of capital would just be unity if capital could be adjusted instantaneously with no cost. In that case the firm would make no super-normal rents.
However, now that we have brought in the consideration that capital cannot be adjusted infinitely fast, in the short run firms will make supernormal rents. In this case the value of the capital of the firm will be unity plus the discounted value of those super-normal rents because capital is slow to adjust.
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7. Microeconomic analysis
investment depends on the path of the future rents, while the rents will also depend on the path of the investment.
For example:
In the short run firms aremaking supernormal rents. To take advantage of those rents capital investment in that industry↑→the rents↓→the investment↓.
At last the industry will approach a steady state in which capital is just being maintained.
q-theory tells that story of interdependence.
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8. Obstfeld’s version of q theory
Assumption:
Output is produced by a constant returns to scale production function Q = F(K, L).
The quantity of labor is fixed. Labor receives a competitive wage, and the return to capital is its marginal product. Labor is fully employed and its supply is fixed at L-bar, so the return to a unit of capital is
There are very high investment costs from investing very fast. Adjustment Costs A=
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9. Introduction to Obstfeld
莫瑞斯·奥博斯特弗尔德(Maurice Obstfeld)毕业于MIT,获经济学博士学位,现任加州大学伯克利分校经济学讲座教授、美国国家经济研究局(NBER)特邀研究员、美国经济政策研究中心(CEPR)研究员和美国计量经济学会(Econometric Society)会员,并且担任IMF、世界银行、欧盟委员会和多国中央银行的顾问。他的研究领域涉及国际金融和宏观经济学。
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10. Deduction
Now the firm’s net cash flow at time s consists of the following:
The firm wants to maximize the present discounted value of these returns subject to the constraint that:
La Grangian expression:
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11. Deduction
We get this from taking the derivative of the LaGrangian with respect to I.
That yields:
The formula says that if the firm is maximizing, investment should proceed to the point where the marginal cost of investment is equal to .
The marginal cost of investment is one unit for the basic capital stock, plus the marginal adjustment cost.
what is the meaning of ？
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12. Deduction
The first-order condition for capital holdings at time s will be given by the derivative of the La Grangian expression with respect to s.
so that ？
This is an investment Euler equation.
It states the firm’s margial condition for investment: The firm is just indifferent between having q (real) dollars, which is the marginal cost of a unit of capital stock.
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13. Deduction
On the reasonable assumption that q must in the long run be equal to one, then we have totally characterized the path of investment and the path of q.
We have one more task here: q really is stock prices. (That q really is stock prices if stock prices reflect fundamentals).
Remember that q is the firm’s internal shadow price of capital.
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14. Deduction Reduplicating what we had before. Is q equal to V /K ?
We start with the Euler equation:
Multiply the RHS and the LHS by K s+1.Convert into F - wL. (Constant returns to scale and labor is paid its marginal product.)
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15. Deduction This yields the equation: We note that by definition:
And we know from our maximization that
Making those substitutions yields the following difference equation:
And that is what we were looking for.
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16. Deduction
This is a first order linear difference equation in the term with forcing function:
Thus: ？
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17. Conclusions So we get exactly what we wanted, which is: q = V /K .
What is the use of this？The use of this is that we now have an interpretation of the investment strategy of this firm.
It makes investments just to the point where the marginal cost of an added unit of investment is the value of its stock per unit of capital.
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18. Summary
Q theory is about the fact that firms will arbitrage. They can get a unit of capital stock by buying on the stock market or by building it.
The marginal cost of an additional unit of the capital stock of the firm should be equated to the value of the stock of the firm per dollar of capital.
If the stock market is low they buy capital stock on the stock market; if the stock market is high then they get capital stock by new investment.
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The end
Thank you!
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