Mathematical Notations that I Never Understand While Reading Articles or Dissertations

Question

I have some problems when I reading mathematical notations. For example:

$$\mathrm{E}_{0}\left\{\sum_{\mathrm{t}=0}^{\infty} \beta^{t}\left[\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)-\int_{0}^{\mathrm{1}} \mathrm{v}\left(\mathrm{h}_{\mathrm{t}}(\mathrm{i}) ; \xi_{\mathrm{t}}\right) \mathrm{di}\right]\right\}$$ This is a optimization equation for representative agent in an economy. This problem here is maximizing this formula. where

\begin{aligned} &\mathrm{E}_{0}= \text{expected value in period 0} \\ &\mathrm{C}_{\mathrm{t}}= \text{ index of household consumption of goods supplied in period t}\\ &\mathrm{P}_{\mathrm{t}}= \text{index of prices of goods supplied in period t }\\ &\mathrm{M}_{\mathrm{t}}= \text{nominal money balances in period t }\\ &\mathrm{h}_{\mathrm{t}}(\mathrm{i})= \text{quantity of labour of type i supplied in period t}\\ &\xi_{t}=\text{ vector of aggregate shocks in period t}\\ &\mathrm{u}(.)=\text { utility in period } t\\ &\mathrm{v}(.)=\text { disutility in period } \mathrm{t}\\ &\beta^{\mathrm{t}}=\text { discount factor for period } \mathrm{t} \end{aligned} 1- First of all, I don't understand the $\xi_{t}$ notation in $\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)$. Does semi colon here mean that this vector notation can affect the equation's behavior exceptionally, but the other variables affect it all the time?. If there would be a positive or negative shock already, then u and v will be effected through the prices and wages. Why should we use this vector here?
2- Why do we need to take the integral of the disutility function, but don't need to take the integral of the utility function?
3- There are no enough explanations for newbies in articles in mathematical economics, and I can't find a textbook for this topic. I studied Varian's Intermediate Microeconomics, but there is no chapter for this. I looked up books that recommended here but again there is no chapter at all. Is there a book that covers all classical mathematical solutions and explains them in detail for economics?

Answer 1

1. The usage of ; is not always standardized so it can depend on source. In this case I presume you took the equation from Interest and Prices: Foundations of a Theory of Monetary Policy By Woodford (or some source that used exactly the same notation), as exactly the same expression appears with all variable names being identical on pp 141 (however, in future you should indicate the source).

   Based on the usage through the book the author uses the ; to separate variables and shock parameters. Hence, it is just to indicate that the $\xi$ is not a variable but given by the shock parameters of the economy.

2. Because the arguments of $u$ are already aggregated. $C_t$ is an aggregate consumption index based on the constant elasticity of substitution aggregator:

$$ C_t \equiv \left( \int_0^1 c_t(i)^{(\theta-1)/ \theta}di \right)^{\theta/(\theta-1)} $$

and the price level $P$ is again already aggregate price index:

$$P_t \equiv \left( \int_0^1 p_t(i)^{1-\theta} di \right)^{1/(1-\theta)}$$

So there is already integral over all goods within the utility. Author just written it in this way to 'clean up' the expression. You can always do that by substitution, you could clean it even more by lets say stating that $V(\cdot) = \int_0^1 v\left( h_t(i) ; \xi_{t} \right) di$ and you could then even write it in single equation without any integral at all (this is really more or less an aesthetic/ trying to make your work easily readable issue).

3. Is there a book that covers all classical mathematical solutions and explains them in detail for economics?

There is no book that covers all mathematical solutions in economics. That would be the longest book ever and you would probably need a truck to move it around. However, there are useful textbooks that cover majority of the most important models in each field.

Intermediate Microeconomics by Varian is undergraduate book so of course it is not on the same level. If you are interested in microeconomics then MWG microeconomic theory or Varian Microeconomic Analysis are the best texts (the former one is quite exhaustive - it is very thick tome).

However, based on the model above I assume you are interested in Macroeconomics. For that Romer Advanced Macroeconomics is a standard graduate handbook. However, even here it would very much depend on exact field you want to study. If you are interested in solely economic growth then Economic Growth by Barro and Sala-i-Martin is much better than Romer which is more general handbook. If you are interested business cycle then the above mentioned Interest and Prices by Woodford is excellent source. If you like monetary theory then Monetary Theory and Policy by Carl E. Walsh is classic text.