Exercises in Mathematcs for NEGB01, Quantitative Methods in Economics. Part 1: Wisniewski Module A and Logic and Proofs in Mathematics

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1 Eercises in Mathematcs for NEGB0, Quantitative Methods in Economics Problems marked with * are more difficult and optional. Part : Wisniewski Module A and Logic and Proofs in Mathematics. The following sets of co-ordinates are two points on a line. Draw the lines in a diagram. i) (, 3) and (-, 0) ii) (0, 0) and (, -3) iii) (5000, 0,5) and (500, -). A company s cost for producing units is given by: b()= *+. Determine i) b(0), ii) b(00), iii) b(0), iv) b(0)-b(00) and v) b(+) 3- A demand function is given by the following: D(p)= 6,4-0,3*p a) Determine i) D(8), ii) D(0) b) If equilibrium demand D(p)= 3,3 what is the equilibrium price? 4. The cost (in millions of SEK) of reducing pollution in a lake by k% is given by: 0* k C(k)= 05 k a) Determine the cost of reducing pollution in the lake by i) 0 % ii) 50% iii) removing ALL pollution from the lake. b) Eplain (in words) the meaning of the epression C(50+h)-C(50). 5. Simplify without using a calculator. i) 6 /4 ii) 5 /7 *5 6/7 iii) (4 8 ) -3/6 6. Simplify without using a calculator. i) (3 +4 ) -/ ii) a*a / *a /4 *a 3/4 iii) ((( 3* ) ) *( * a a ) ) 3 a 7. a) Solve for K. 3*K -/ *L /3 = /5 b) Solve for a**(a*+b) -/3 +(a*+b) /3 =0 8.

2 K(t) is the capital stock a firm has at the end of year t. Assume that K(t) increases by p% each year. If K(0)=50; what is: i) K(), ii) K(), iii) K(q), vi) K(0) 9. Change and to implication and equivalence signs where appropriate. a) =9 and =3 b) + > 3 and > c) Peter is taller than Paul and Mary is taller than Peter and Mary is taller than Paul d) It is a warm summer day and we are going swimming e) 4 =5 and = 5 0. Are these two statements true or false? a) is any real number and / is the square root of b) is any real number and /3 is the third root of. Epress the following in terms of ln 3 a) ln 9 b) ln ( 3 ) c) ln ( 5 3 ) d) ln 8. Find when a) 3 =8 b) ln =3 c) ln ( -4+5)=0 d) =8 e) 3ln +ln = 6 f) = Solve for t. a) =e at+b b) e -at = 0,5 4. Show that: a) ln- = ln(*e - ) b) ln -ln y+ln z = ln z y c) 3+ln =ln( e 3 ) 5. True or False? a) ln 5-ln 0 = -ln b) (ln A) 4 = 4 ln A c) ln B= ln B A B ln A ln A ln B lnc d) ln C e) ln B lnc ln A(BC) - 6. a) Show that if f()=00* ; then f(t*)=t *f() for all t. b) Show that if P()= 0,5, then P(t*)= t 0,5 *P(). Part Wisniewski Module B and Equations and systems of linear equations. Which of the following functions are linear? Identify the slope and intercept of those functions that are linear. Sketch the functions roughly using this information. i) f()=*+3 ii) h(g)=4-g/ iii) y=0,5+*-3* iv) z=4 +0,5 v) T(s)=s vi) L=0,5p+45. Solve the following systems of linear equations. Do a) and b) both by substitution and elimination.

3 a) + 3y = () 5 y = () b) y =8 () 0 + 5y = 65 () c) 5 + y + 3z = 6 () y 3z =3 () 7 + 4y + z = -4 (3) 3. Two linear functions, L() and M() cut each other at point P. L passes through the co-ordinates (,) and has a slope = 3; M passes through the co-ordinates (-, ) and (3, -). Determine the equations of L and M as well as the co-ordinates for P. (Give an eact answer for coordinates of P.) 4. A national income model is Y = C + I + G C = a +b(y-t) a>0 0<b<0.90 I = d+ry d<0 0<r<0.0 G, r and T are eogenous r = the rate of interest, other notation is as in Wisniewski a) Find the public consumption multiplicator. Interpret it. b) Find the ta multiplicator. Compare with a). Interpret! c) Find the public consumption multiplicator if T = ty 0<t<. Compare with a) and interpret. Part 3 Wisniewski Module C.-C.3 and supplement Roots of polynomials and the factor theorem Part 3. Roots of polynomials. Find the roots of the following quadratic equations and write the left sides of the equations as products of linear factorsi) +5+6=0 ii) 0* 4 =0 6 iii) -8+8=0 iv) =0. a) Find the zeroes (roots) of p() = b) One of the zeroes of the polynomial p() = 3 +3/ 9 +4 is =. Use the factor theorem to find the other two. (Hint: Calculations are easier if you write numbers as fractions rather than with decimals.) Part 3. Differentiation 3. Differentiate the following epressions: a) y=5 b) y= 4 c) y=9 0 d) y= 7 4. Differentiate the following epressions. Assume g () is known. a) y=- 6 *g()+8 3

4 b) y= g( ) Differentiate the following epressions: a) s= b) f()= c) B()= * d) f(a)= A * A e) g(a)= *a+h+ a f ( a ) h is a constant. h 6. Differentiate the following epressions: a) f()= +4 b) f()=+ -0,6 c) f()= d) f()= e) f()= Differentiate the following epressions using the product rule. a) y= ( -)( 4 -) b) z(t)= ( t )*( t 5 ) t c) h= ( )* 5 8. Differentiate using the quotient rule. f()= 9. Differentiate S(p)= p*d(p) where D is a differentiable function of p 0. Determine the values of where dy d 0 a) f()= b) f()= 0,5( 4-6 ) 3 c) f()=. Determine the equations of the tangents to the following functions at the given points. a) y= 3-- at = 4

5 b) y= at = If you found a) and b) difficult, do c) and d) too, otherwise skip them. c) y= * at = d) y= 4 * 3 at =0. Differentiate: at b a) y(t)= ct d b) y(t)= at bt c y 8 c) L(y)= y 3.* Show that if f ( ) = -n then df ( ) = -n -n- using the quotient rule. d 4. Differentiate: a) g= (+) 3 b) h= (-) 5 c) R= ( 3 4) d) F= ( 3 ) e)* B= / f) y= (- ) 33 g) d(t)= (at +) -3 h) y(t)= (at+b) n 5. Determine dy d a) +f() b) (f()) 4 c) *f() f ( ) d) when y= 5

6 6. Determine dy d using only in the epression. If y= 5u4 and u= + 7. Determine dy dt 8. Determine dk dt 9*. Determine dy d 5 a) ( ) b) c) a (p+q) b as a function of t if y= -3*(v+) 5 and v=t 3 /3 as a function of t if K=AL a and L=bt+c. when y= 0. If a(t) and b(t) are both differentiable functions and A, L and B are constants then determine ( t) ( t) a) = (a(t)) *b(t) b) =A(a(t)) L *(b(t)) B. Differentiate the following epressions: a) C()=0q -4q(5-0,5) / b)* F()= f( n g()). Find the second order derivatives of the following functions: a) y= b) y= c) y=(+ ) 0 3 a) z= 0t- t 3 3 determine d 3 z 3 dt 4. a) f(z)=z -4 determine f (4) (). t b) g(t)= t determine g () In the following, note that e f() can be written as ep(f()) 5. Differentiate the following epressions where f() = a) e -3 b) e 3 c) e / d) 5ep( -3+) 6. Find the derivatives of the following functions: a) f() = ep(e ) b) f(t) = e t/ +e -t/ c) f ( ) e e d) g(z)=(ep(z 3 )-) /3 e) h() = e 7. Differentiate the following functions: 6

7 a) y= ln(+) b) y= ln+ c) y= ln d) y= ln e) y= ln( +3-) 8. A function is given by y=ln. Find the equation of the tangents to the graph at the points: a) = b) =e 9. A function is given by y=ln. Find the equation of the tangents to the graph at the points: a) = b) =e 30*.. Some derivatives f () are easiest to calculate if instead of f() you differentiate g() = ln f() and use the fact that ( ) ( ). Use this method (logarithmic differentiation) to find the derivatives of the following functions: a) y= / 3 ( ) b) y= c) y= (-) / ( +)( 4 +6) Part 3.3 Elasticities 3. Determine the general elasticity for the following functions where f()= a) 3-3 b) c) 0,5 A d) h)a*g() i) + j) a+ b k) a+b/ e) 3 f) + g) (- ) 0 3.* A firm s total cost function is given by C(Q) where Q is the quantity produced. Average cost is given by AC(Q) = C ( Q) Q Show that AC (Q)=0 if and only if the elasticity of C with respect to Q, =. Part 4: Optimization of functions of one variable. Wisniewski Module C4-C6 If you are short of time, save 5) and 7) for later revision..find the stationary points of the following functions. Determine the nature of these points (ma, min, inflection?) Sketch the functions roughly using this information. a) f()= b) f()= c) h(y)=y 5-5y 3 d) z()= (-) 5 (+) 4. A function is given by: F()= 3 5 where, a) Show that ( ) ( ) b) Show that F() is decreasing for all values of. 7

8 3. Determine the stationary points of the function G()= /3 *(-7) and determine the nature of these points. 4. A Cobb-Douglas function has the form. Q( K, L )= AK L a) Define L as a function of K for an arbitrary fied value of Q ; Give this function an economic interpretation. b) Show that L is a decreasing function of K for all K. 5. Determine the stationary points of the following functions. Determine the nature of these points, as well. a) y= -,05+,06-0,04 8 b) h()= A square piece of cardboard has the dimensions 8 X 8 cm. If we cut out 4 identical squares from each corner of the cardboard, it can be folded to form an open bo. The side of each cut corner is denoted. a) Determine an epression V(X) for the volume of the bo 8 cm b) For which value of is the volume of the bo maimized? c) For what values is V() increasing, decreasing? 7. a) Show that if f()= 3-(-) then f() 3 for all. b) Find the stationary point of the function T= -(-) / 8 cm 8. A company in a perfectly competitive market has total costs TC=Q +0Q+900. where Q is quantity of the good produced. The price of the product is constant and =80 kr. a) Determine the profit function for the company. b) At what Q is profit =0? c) At what Q is profit maimized? 9). Find the ma. and min points of: a) p()=a+k(-e -c ) where a, k and c>0 b) y= e - Part 5: Functions of more than one variable. Wisniewski Modules A3.6, D. and D.. and supplement Total derivatives. a) z(,y)= y determine i) z(0,) ii) z(-,) iii) z(a,a) b) f(,y)= 3 -y+y 3 determine i) f(,) ii) f(-,3). If F(K,L)= 0K / L /3 where K and L 0 then determine 8

9 a) F( K,L) *b) Determine so that F(tK,tL)= t F(K,L). Does this production function have diminishing, constant or increasing returns to scale? 3. Differentiate the following functions with respect to and y. a) z= +3y b) K=y c) P= 5 4 y -y 5 4. Find the partial derivatives of the following functions. a) F(,y,z)= 3yz+ y-z 3 b) Y(K,L)= 6. Let z(, y) = -8y y 3 a) Find the total differential of z K L al bk 3 3 c) T= ( y 4y ) ( y ) b) if (t) = 3t and y(t) = -t, what is the total derivative of t with respect to t? c) Use substitution to write z as a function of t, differentiate and check that you get the same answer as in b). 7. A firm has the production function Q = Q(L, K) = 0K /4 L / a) How much does the firm produce if K = 65 and L = 64? (Try to calculate without using a calculator.) b) Use the total differential to find a approimate values of Q(64.05; 65.) and Q(00, 75) c) Use a calculator or ecel to find Q(64.05;5.0) and Q(00, 75) rounded to 5 decimals. Part 6 Unconstrained and constrained optimisation of functions of several variables. Wisniewski Modules D3 and D4 Part 6. Unconstrained optimisation. Determine and solve the First Order Conditions (FOC) for the following functions: a) f(,y)= - -y +4+4y-3 b) P(,y)= - -y ++8y-0.. f(,y)= +y -6+8y+35. a) Solve the FOC of the function. b) Show that f(,y) can be written as f(,y)=0+(-3) +(y+4). c) Show that f(,y) 0 for all and y. What does this imply with regard to your answer in a) 9

10 3. A firm has the production function F(K,L)= 80-(K-3) -(L-6) -(K-3)(L-6) where K represents capital and L labour. The price of the product is p=, the cost of capital is r =0,65 and the wage rate is w=,. Find the values of K and L that satisfy: df dk r and p df dl w p. Give an economic interpretation 4. Find the ma., min or saddle points for the following functions: a) f(,y) = 3 +y 3-6y. b) h(,y)= y 3 ( 6--y ) +5, where, y >0. c) f(,y)= 4 +y -y d) g(,y) = ln( + y) 5*. Find the values of a, b and c such that F(,y) = a y+by+y +c has a local minimum at the point (/3, /3) and F(/3, /3)= -/9. Part 6. Constrained optimization 6. Maimize f(,y)= +y given that g(,y)= +y= a) Using substitution. b) Using the relation f f y g g y c) Using the Lagrange method. Interpret the Lagrange multiplier. 7. a) Use the Lagrange method to find the points that satisfy the first-order conditions for a maimum or minimum of the function f(,y)= 3y under the constraint that g(,y)= +y =8. b) Try to illustrate the problem graphically. Can you determine which of these points maimizes / minimizes the function. 8. Use the substitution method to determine the minimum of +y given that +y=a. 9. Henrik s utility function is given by U(,y)= 0 / y /3. The price of good X is 3 kr while the price of good Y is kr. Assume that Henrik will spend 50 kr on goods X and Y. Assuming that Henrik is rational, what quantities of X and Y will he consume? 0

11 0.*. Bengt s utility function is given by U(,y)= 00-e - -e -y. The price of good X is p kr, while the price of good Y is q kr. Bengt will spend R kr on goods X and Y. Assuming that he is rational, determine an epression for Bengt s demand of the two goods.. A utility function is given by U(,y)= 3 +y 3. The price of good X is 9 kr, while the price of good Y is 4 kr. Determine the optimal combination of and y in order to obtain utility U= A Cobb Douglas production function Q(K,L)= 80K 0,75 L 0,5 ; the price of capital is 3, while the price of labour is. Minimize the cost of producing 6400 units. 3. Maimize F(,y)= ln(y ) given that +3y=8.