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a) Question (1): Derive the general formula of the Euler equation using the lagrangian.

b) Question (2): We have discussed in one of the first classes, what consumption smoothing means, but if you don't remember look it up. What does the Euler equation imply about consumption smoothing?
Consumption smoothing is the economic concept used to express the desire of people to have a stable path of consumption. People desire to translate their consumption from periods of high income to periods of low income to obtain more stability and predictability.
The consumption euler equation implies that individuals should to smooth consumption over time.

c) Question (3): Derive the intertemporal budget constraint.
Two period budget constraints:

Where the superscripts d and s reflect demand and supply decisions, respectively. The household takes as given W, R, and . Note that S may be positive or negative. Negative values of S imply borrowing. Implicit in our formulation, however, is the assumption of perfect capital markets.
Combine the two period budget constraints into a single intertemporal budget constraint:

d) Question (4): The trade balance of an economy can be defined as TBt = Yt � Ct, while the current account is the trade balance minus the interest payment on debt, that is CAt = TBt � rDt�1. Express the current account in period t as a function of Yt. Don't forget that in period t the household may have debt!.

e) Question (5): What do these results imply about the cyclicality of the current account? (That is, what is the sign of the derivative of the current account with respect to the income? If you don't know what the terms procyclical and countercyclical mean, look it up!)
The current account is countercyclical.

f) Question (6): What can you say about the relative variance of the consumption? (That is, is it more volatile than the output or not. Take the derivative of the consumption with respect to the output, and if it's larger than one, it's more volatile, if it's lower than one, it's less volatile.)

g) Question (7): Define a small open economy. Is Hungary a small open economy?
A small open economy, abbreviated to SOE, is an economy that participates in international trade, but is small enough compared to its trading partners that its policies do not alter world prices, interest rates, or incomes.
Hungary is a small open economy.

h) Question (8): Download data on the current account and the GDP of Hungary, and save it in a csv file (which contains only these variables between 2000 and 2016). Submit this csv file. Data can be found at the website of the Central Statistical Office (click here). Use table 7.3.1 for the GDP. For the current account use table 7.3.19.1. Notice that they are in different units! Make sure that in your data file you use the same units!