# Grodal on Macroeconomics

Birgit Grodal, Comment on L. H. Summers, The Scientific Illusion in Empirical Macroeconomics. Scandinavian Journal of Economics 93 (2), 155-159, 1991.

Summary

According to Grodal, the reason why sophisticated macro-econometrics has little impact is that the underlying macroeconomic theory is insufficiently developed.

Macroeconomic models are based on unwarranted simplifications. They use one or a few commodities, one or a few representative consumers and producers, perhaps a public sector, and simple institutional arrangements. Yet the conclusions drawn from these models are treated as if they hold in economies with many interacting agents.

There is no basis in economic theory for believing that these models should give a good description of the way an economy with many agents operates.

Only in economies where all consumers have identically homothetic preferences can the demand side can be represented by a representative consumer.

That is true if all income distributions are allowed. If only a fixed income distribution is considered, then it may be possible to represent the demand side by a representative consumer, but only under severe restrictions on the preferences of the individual agents. In this case, the welfare implications of economic policy for the representative consumer can be the opposite of the welfare implications for all the original consumers.

Thus it is not surprising that macroeconomic relations derived from theory are usually rejected empirically. The only way to obtain better empirical macroeconomic models is to develop better macroeconomic theory.

Grodal recommends studying the distribution of agents’ characteristics and using it to derive conclusions about aggregate behavior.

# University of Copenhagen Gold Medals 1940–1980

Recipients of the University of Copenhagen’s Gold Medal in Mathematics and Economics for the years 1940–1980.

 Year Mathematics Economics (“Statsvidenskab”) 1980 Lars Tyge Nielsen None 1979 None Lars Tyge Nielsen 1978 None None 1977 Jesper Laub None 1976 None None 1975 None None 1974 Susanne Aref None 1973 Dorte Olesen None 1972 None None 1971 None None 1970 Søren Jøndrup, Knud Lønsted None 1969 Christian Berg Birgit Grodal 1968 None None 1967 None None 1966 Søren Johansen None 1965 None None 1964 Arne Brøndsted Erling B Andersen 1963 None None 1962 None None 1961 None None 1960 None None 1959 None None 1958 None None 1957 None None 1956 None None 1955 None None 1954 None None 1953 Frans Handest None 1952 None None 1951 None None 1950 Sigurdur Helgason None 1949 None None 1948 None None 1947 None None 1946 None None 1945 Ole Rindung None 1944 None Bent Hansen 1943 None None 1942 Thøger Bang None 1941 Helge A. B. Clausen Hans Julius Brems 1940 None None

# MathJax

I have just discovered MathJax and installed the wordpress plugin MathJax-LaTeX as a replacement for WP-LaTeX. The point is to typeset mathematics within html-pages.

Here is an example of how MathJax renders mathematics:

Given $$(\mu,\sigma)$$ and a setup $$(\Omega,\mathcal{F},P,F,W)$$, a solution on $$(\Omega,\mathcal{F},P,F,W)$$ of the SDE $$(\mu,\sigma)$$ is an $$N$$ dimensional Itô process $$X$$ on $$(\Omega,\mathcal{F},P,F,W)$$ with a potentially random initial value $$X(0)$$, such that the process $$\mu(X,t)$$ belongs to $$\mathcal{L}^{1}$$, the process $$\sigma(X,t)$$ belongs to $$\mathcal{L}^{2}$$, and for all $$t \in [0,\infty)$$,
$$X(t) = X(0) + \int_{0}^{t} \mu(X,s) \, ds + \int_{0}^{t} \sigma(X,s) \, dW$$

# N(d1) and N(d2)

No, N(d1) is not the probability of exercise.

In one of my classes I derived the formulas for the values of standard options and various digital options in the Black-Scholes models, the point being to illustrate various concepts – the state price process, risk-adjusted probabilities, and the use of different numeraires.

We got into an argument about the meaning of N(d1) and N(d2).

I published a paper about this a number of years ago in Revue Finance, the journal of the French finance association: “Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model,” Revue Finance (Journal of the French Finance Association) 14 (1993), 95-106. [Abstract][Abstract on the journal's website][Paper (pdf)]

The paper explains N(d1) and N(d2) and relates them to the single-period and multi-period binomial models.

After so many years, people are still puzzling over this.

What is N(d1)? Well, N(d2) is the risk-adjusted probability of exercise. N(d1) is something else: It is the factor by which the present value of contingent receipt of the stock (contingent upon exercise) falls short of the current stock price.

# Mathematical Genealogy

My Ph.D. is in economics, not in mathematics, but I did write a master’s thesis in differential topology, some of which was published in: Transversality and the Inverse Image of a Submanifold with Corners. Mathematica Scandinavica 49 (1981), 211-221.

My thesis advisor was Vagn Lundsgaard Hansen, then at the University of Copenhagen, now at the Technical University of Denmark. Through him, I trace my mathematical ancestry back to Birkhoff, and possibly all the way back to Poisson, Lagrange, Euler, Johan and Jacob Bernoulli, and, finally, Leibnitz.

Apart from being my advisor, Lundsgaard Hansen was also my teacher in first-year undergraduate mathematical analysis. The course started with an unforgettably vivid exposition of the topology of metric spaces — pulling back open and closed sets and mapping compact sets forward and so on. Later on I took his course in algebraic topology, where the main agenda was to use category theory to translate topological questions into easier algebraic ones.

Category theory was apparently invented by Samuel Eilenberg together with Saunders Mac Lane.

Samuel Eilenberg

This bust of Samuel Eilenberg stands in the corner of the Faculty Lounge in the Columbia Mathematics Department. Eilenberg was a professor in the department from 1947 and until his retirement in 1982.