Mathematical Economics

MME Mathematical Economics ……….

Program

Second year, winter semester ………..

Year (winter/summer semester)

Mathematical Economics Name of the course Wioletta Nowak The Professor

Biography of the Professor

Wioletta Nowak is an assistant professor at the Institute of Economic Sciences at the Faculty of Law, Administration and Economics of the University of Wroclaw, Poland. She has a MSc in theoretical physics and a MA in philosophy from University of Wroclaw, and PhD in economics from the former Higher School of Economics in Wroclaw (Wroclaw University of Economics today). Her research interests span mathematical economics, economic growth, economic development and international economics. She is (co)author of 5 books and over 60 papers.

Requirements for passing a course:

Written exam, five exercises based on exercises from the list below, to pass a student needs 50%, for B – 70% and 90% for A.

Exam exercises:

1. Solve the following utility maximization problem

a)





8 2 2 1 2 1 max 2 1 2 1 ,₂ 1          _ x x x x x x _b) 9 3 1 max 2 1 2 2 1 1 , ₂ 1      x x x x x x x x c) 5 4 3 max 2 1 4 , 0 2 6 , 0 1 ,2 1   x x x x x x _d)







I x p x p x x x x     2 2 1 1 2 1 , 3 1 2 max 2 1 , , 0 2 1 p I  p .

2. Find the demanded bundle for a consumer whose utility function and budget constraint are the following

b)





2



4



2 1 , 2 1 2 1        _  x x x x u , 2x₁ x₂ 10 c)





3



3



3 1 , 2 1 2 1        _  x x x x u , 2x₁x₂ 10 d)



 



      _   2 4 1 2 , _{2 1 2} 1 x x x x u , x12x2 20 e)





4 1 2 2 1 1 2 1,x x x x u  , 5 3 3 1 2 1  x  x f)





2 ₂ 1 1 2 1,x x x x u  , x₁ 2x₂ 6 g) u



x₁,x₂



x₁x₂, 2x₁x₂ 8 h) u



x₁,x₂



4x₁x₂, 2x1x2 8

3. Find the demand functions x₁



p₁,p₂,I



and x₂



p₂,p₁,I



if the budget constraint is I

x p x

p_{1 1}  _{2 2}  , p₁,p₂,I 0 and the utility function is given by

a)







2



1



3 1 , _{2 1 2} 1 x  x  x  x u , b)





2 1 2 4 1 1 2 1,x 3x x x u  , c) u



x₁,x₂



 x₁x₂3, d) u



x₁,x₂



alnx₁(1a)lnx₂, a(0,1), e) u



x₁,x₂



 x₁2 x₂2, f)





3 2 1 2 2 1 1 2 1,         x x x x u , g) u(x₁,x₂)2min



4x₁x₂,x₁4x₂



, h) u x x



₁, ₂



min



4x₁ x x₂, ₁7x₂



. 4. Draw the indifference curve:

a) _{1 2} min



₁,2 ₂



2 1 ) , (x x x x u  , u



x₁,x₂



4;

b) u(x₁,x₂)min



3x₁x₂, x₁3x₂



passing through point

 

4,4 ; c) u



x₁,x₂



min



x₁7x₂,4x₁x₂



, u



x₁,x₂



9.

5. Solve the expenditure minimization problem

a) 3 1 2 3 1 1 2 2 1 1 , ₂ 1 min x x u x p x p x x   b) 2 1 2 4 1 1 2 2 1 1 ,₂ 1 min x x u x p x p x x   0 , , ₂ 1 p u  p .

6. The utility function is given by

A) u



x₁,x₂



x₁2 x₂2; B)





3 1 2 3 1 1 2 1,x x x x u   ; C) u



x₁,x₂



 x₁n x₂n; D)



_{1 2}



₁ ln ₂ 4 1 ln 2 1 ,x x x x u   ; E) u x x



_{1 2}



x₁n xn n 1 2 1 0 ,   ,  .

a) 

 

p I,  f p v p I



,

 

,



– the Marshallian demand at income I is the same as the Hicksian demand at utility vv

 

p,I

b) f

 

p,u 



p,e

 

p,u



– the Hicksian demand at utility u is the same as the Marshallian demand at income e

 

p,u

c) e p v p I



,

 

,



 I – the minimum expenditure to reach utility uv

 

p,I is I d) v p e p u



,

 

,



u – the maximum utility from income e

 

p,u is u

e) 

 

 

 

 

      i j i j i j p I p f p u p p I I p I , , , ,

   , i, j1,2 i j – the Slutsky equation 7. Check the returns to scale for the following technologies

a) f



x₁,x₂



3x₁x₂, b) f



x₁,x₂



 x₁2x₂ , c) f x x



₁, ₂



 x x₁1 4/ ₂3 4/ , d) f



x₁,x₂



 x₁2x₂3, e) f



x₁,x₂







x₁1/4 x1₂/4



4, f) f x x



₁, ₂



 x₁ x₂2. 8. For the following technologies

A)













 1 2 1 1 a x ax A y    , B)













 1 2 1 1    _{ }  Aax a x y , 0   1, 0 a 1, A0 C) y Ax₁cx₂d, A,c.d 0 compute:

a) the marginal product of capital and marginal product of labour, b) the technical rate of substitution,

c) the output elasticity of capital and output elasticity of labour, d) the elasticity of substitution,

e) the elasticity of scale, f) y

0 lim



 .

9. A firm has a production function given by

A) 3 1 2 3 1 1 4x x y  ; B) 2 1 2 4 1 1 3x x y ; C) 2 1 2 3 1 1 5x x y ; D) 3 1 2 6 1 1 12x x y  .

a) What are the factor demand functions?

b) What are the conditional factor demand functions? c) What is the cost function?

d) What is the supply function? e) What is the profit function?

(Do calculations for long and short run).

10. For the technologies

       1,2 2 4 1 min x x y and 1 ₂ 2 x x y  compute: a) the total cost and the cost of production of 1 unit of output,

b) the marginal and average cost. Assume that prices of inputs are (4, 6).

11. A monopoly has a production function y x x



_{1 2}



x x₁ 1 2

2 1 3

a) p y( )6y 1 2_{, v} 1 2, v2 3, b) p2 , v x₁

 

₁ 2x₁, v x₂

 

₂ 3x₂, c) p y( )6y 1 2_{, v x}

 

_x 1 1 2 1, v x2

 

2 3x2.

12. Suppose that we have two firms that face linear demand curve p2001 y  y

2( 1 2) and their cost functions are c y_{1 1} 1 y₁2

2

( ) , c y₂( )₂ 10y₂, respectively.

a) Compute the Cournot equilibrium amount of output for each firm and firms’ profits.

b) If firm 2 behaves as a follower and firm 1 behaves as a leader, compute the Stackelberg equilibrium amount of output for each firm and firms’ profits.

Please repeat calculations if: A) p401 y  y 2( 1 2); c y1 1 y1 2 2 ( ) ; c y_{2 2} 1 y₂2 2 ( ) ; B) p30 3 (y₁ y₂); c y₁( )₁ 3y₁; c y₂( ₂) y₂. C) p1002(y1 y2), c1(y1)2y1, 2 2 2 2 4 1 ) (y y c  . D) p40(y1y2), 2 1 1 1 4 1 ) (y y c  , c2(y2)3y2.

13. The traders’ utilities are given by u x x1



₁, ₂



 x x_{1 2}2 and u x x2



₁, ₂



 x x₁1 2 ₂1 2. Their initial endowments are the following a1 ( , ) and a2 2 2 ( , ) . Traders come to a market and 4 4 exchange commodities to maximize their utilities. Compute the price vector in equilibrium. Compare the utilities before and after the exchange.

Please repeat calculations if:

A) a1 (1,4), a2 (2,1), u1



x₁,x₂



 x₁3x₂,





4 3 2 2 3 1 2 1 2 ,x x x x u  . B) a1 ( , ) , a10 10 2 ( , ) , 20 5 u1



x₁,x₂



x₁23x₂13, u x x2



₁, ₂



x x₁1 3 ₂1 2. C) a1 ( , ) , a3 9 2 ( , ) , 1 3 u x x1



₁, ₂



 x x_{1 2}1 3, u x x2



₁, ₂



x x₁3 ₂. Sources: