平成 26 年度実施
東北大学大学院情報科学研究科 博士課程前期・入学試験問題 (2015 年 3 月 3 日)
専門試験科目群第 7・社会科学群

Problem E–1 Let the utility function of consumers be U (x1 , x2 ) = 1 − e−x1 − e−x2 , theprice of good 1 be p and the price of good 2 be 1.
(1) Given income of I, derive the Marshallian demands (ordinary demends) for two goods
and the indirect utility function V (p, I).
(2) Given utility level u, derive the Hicksian demand function (compensated demand function). Find the expenditure function M (p, u).

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平成 26 年度実施
東北大学大学院情報科学研究科 博士課程前期・入学試験問題 (2015 年 3 月 3 日)
専門試験科目群第 7・社会科学群

Problem E–2(1) Explain Zipf’s law (or rank-size rule) and its application in the contextof regional science.
(2) Explain the distinction between the “marketing” and “transport” principles in Christaller’scentral place theory. Drawing figures might be of help.

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平成 26 年度実施
東北大学大学院情報科学研究科 博士課程前期・入学試験問題 (2015 年 3 月 3 日)
専門試験科目群第 7・社会科学群

Problem E–3

Consider a closed economy that is completed with two countries and pro-duces a single (composite) good. Then the system of output equations in physical units (e.g.
tons or litters) for the inter-regional physical table can be written as follows:a11 x1 + a12 x2 + y11 + y12 = x1 a21 x1 + a22 x2 + y21 + y22 = x2 , 
where xs and yrs denote the output of country s and the amount of country r’s productthat is finally consumed in country s, respectively. ars represents the amount of country
r’s product that is required to produce one unit of goodin country s, and let A denote the a11 a12 matrix composed from those physical input coeﬃcients .
a21 a22(1) Given ars and yrs , calculate the output x1 of country 1. For the output system being
economically meaningful, what characteristics the input coeﬃcient matrix A should possess?
(2) Let pr denote the price of good produced in country r. Then write the system of output
equations in monetary terms by assuming that the transport costs between countries arenegligible.
(3) For convenience, we call the currency of country 1 as “dollar”, and that of country 2 as“Euro”. Note that we have p1 = µp2 when 1 Euro is equivalent to µ dollars. Then provide
the formula for the trade balance T B of country 1.
(4) When the exchange rate µ is determined so as to keep the balance of payments to zero,
what terms should be added to T B obtained above?

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平成 26 年度実施
東北大学大学院情報科学研究科 博士課程前期・入学試験問題 (2015 年 3 月 3 日)
専門試験科目群第 7・社会科学群

Problem E–4

For function f (x), let′d ln f (x)f (x)fˆ(x) ≡=.
dxf (x)Given functions f1 (x), f2 (x),(1) derive f\
1 + f2 ,fd
1 f2 ,(\f1 );f2(2) calculate ĝ(x), where g(x) is the following determinant:f1 (x)
g(x) =f2 (x) − f1 (x)6

f1 (x) −1 2f2 (x) − f1 (x) .
f1 (x)

3f2 (x) − f1 (x)34平成 26 年度実施
東北大学大学院情報科学研究科 博士課程前期・入学試験問題 (2015 年 3 月 3 日)
専門試験科目群第 7・社会科学群

Problem E–5

The demand curve D and the supply curve

S are generally depicted as the functions of price p as shownin the figure. Consider the following system of linear models:D = α0 − α1 p + u . . . (a)S = β0 + β1 p + v, 
where u and v are the disturbances in respective equations.
(1) Suppose the parameters in qt = α0 − α1 pt + ut are estimated from the time series dataof demands q and prices p. The result can be called neither demand function nor supplyfunction. Draw a figure to explain the reason.
(2) Regarding (a) as the simultaneous equations, and discuss their identification problem.
Let g denote the number of endogenous variables in an equation, and k denote the number ofsystem variables (both endogenous and exogenous) that are not appearing in that equation.
Examine the order condition (= the necessary condition for the parameters being able to beestimated) for (a).
(Hint) When k = g − 1, an equation is called “just-identified”.
(3) How the equations in (a) should be modified in order to properly estimate the demand andsupply functions? Propose the modification that includes addition of appropriate variables.

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