3
A delay of two years, (1928 to 1930) in the publication by Keynes, then editor of the Economic Journal of Harrod's original "marginal revenue curve" denied the Oxford economist the primacy in this field. This sad turn of circumstance was to repeat itself several times in Harrod's career. Another independent discovery by Harrod (1931), effectively the long-run envelope of short-run average cost curves, also went unrecognized - the credit being awarded to Viner. In that same article, he laid out the analytical foundations for the theory of imperfect competition - but Joan Robinson took the prize. The laurels for his remarkable multiplieraccelerator model, developed in his Trade Cycle (1936), were given to the mathematically-expressed versions by Samuelson and Hicks.
∫
dY / dt = (1 / s ) ∫ dI / dt
1 Y + c3 = I + c 4 s 1 1 Y = I + (c3 + c 4 ) = I + B s s 1 Yt = I t + B s
把
I t = I 0 e ( s / v )t
代入上式，有：
16
( )t 1 Yt = ( ) I 0 e v + B s
哈罗德—多马经济增长模型 多马经济增长模型： 第六章 哈罗德 多马经济增长模型： 储蓄对增长的贡献 经济增长研究重新出现的背景：
1、1930年代的世界经济大危机 2、战后的经济萧条 3、柯布和道格拉斯生产函数的提出 4、凯恩斯的收入决定理论
1
关于作者：
罗伊·哈罗德（Roy F. Harrod ；1900—1978） 1924年开始在牛津大学教授经济学，直到 1967年退休。曾在英国国家统计局任职，第 二次世界大战期间担任丘吉尔顾问。 多马（Evsey D. Domar；1914-1997）生 于波兰，1936年移民到美国。1947年获得哈 佛大学博士。曾经在the Carnegie Institute of Technology、芝加哥大学和霍布金斯大学 任教。1958起为麻省理工学院终生教授。 1984年退休。
14
给定初始条件为：
I0 = e
有：
c1 + c2
I t = I 0 e ( s / v )t
（12）
方程（12）给出了为了实现投资充分使用的投资增长路径。它显示了投 资必须以一个有保障的比率增加，这就是s v ，经济才能保持均衡增长。
15
投资增长率与产出增长率是一致的，为了显示这一点，我们对方程（10）两边积分：
5
His contributions to international trade (1933, 1958) and imperfect competition (1933, 1934, 1952) have also been given latter-day recognition. His less rigorous pieces on economic policy (1963, 1965, 1968, 1969) were also remarkable. He was appointed a Reader in Economics at Nuffield College, Oxford in 1952, and is remembered in a Fellowship in Applied Economics at that College. Outside economic theory Harrod's claim to prominence was his work on inductive logic (1956), his role on the Statistical Staff and as personal adviser to Winston Churchill during WW II, and his unofficial advice to Harold MacMillan as Prime Minister. He also devoted great energy to the life of his Oxford college.
s
（13）
设 有：
1 Y0 = ( ) I 0 , s
I 0 = sY0
代入方程（13），当t=0时，得到：
1 Y0 = ( ) I 0 + B = Y0 + B s
即B=0
17
因此有：
Yt = Y0 e
4
The equations of the IS-LM model were written down by Harrod (1937), but the (later) drawing of the diagram by Hicks robbed him of his claims to precedence. His 1939 paper on entrepreneurial behavior one of the first statements of the idea that there are natural selection or "evolutionary" mechanisms towards profit-maximizing behavior was largely ignored and left for Alchian (1950) to propose. Finally, another discovery gave him some name recognition: Harrod's (1939) "Essay in Dynamic Theory". The idea, which marked the beginning of the modern theory of growth, had been followed up by Evsey Domar, but at least he got his name on the model this time: the "HarrodDomar Model". In his 1948 book, Towards a Dynamic Economics, as well as in a series of essays (1960, 1963, 1975) he developed this further, highlighting the instability problem of this model and launching the entire post-war research program on economic growth - and, indeed, reviving business cycle theory as well.
10
2．多马模型 ．
从总供给的角度看，潜在总产出的增长决定于两个因素， 一个是投资量，另一个是潜在的社会平均投资生产力，即：
dY pσ = Y K ——潜在的社会平均投资生产力
代入（7），得
——潜在总产出
将
σ = Y / K = 1/ v
dY p dt = 1/ v ⋅ I
（8）
11
Y =C 从总需求的角度看，按照凯恩斯理论，总需求为 + I 式求时间的全微分 ：
，对等
dY dC dI cdY dI (1 − s ) dY dI = + = + = + dt dt dt dt dt dt dt
（9）
整理后有：
dY (1 / s ) dI = dt dt
（10）
12
在均衡状态下，总供给等于总需求，有：
2
Roy Harrod taught economics, and produced his original contributions to the subject, at Oxford between 1924 and his retirement in 1967. Elected to a Lectureship at Christ Church Oxford in 1922, he then spent a few months at Cambridge with J.M. Keynes, with whom he remained in close touch for the rest of Keynes' life, and whose official biography he published in 1951. With Hicks and Meade he was among the Oxford economists included in Keynes' circle of correspondents. A founder of the Oxford Economics Research Group along with Hubert Henderson and others, his unique contribution in urging a "dynamic" rather than "static" approach to economic issues was an Oxford contribution, accepted only later among the Cambridge economists.