Solutions to selected exercises fromJehle and Reny (2001): AdvancedMicroeconomic TheoryThomas HerzfeldSeptember 2010Contents1 Mathematical Appendix21.1 Chapter A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Chapter A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Consumer Theory122.1 Preferences and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The Consumers Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Indirect Utility and Expenditure . . . . . . . . . . . . . . . . . . . . . . . 162.4 Properties of Consumer Demand . . . . . . . . . . . . . . . . . . . . . . 182.5 Equilibrium and Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Producer Theory233.1 Pro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Duality in production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 The competitive firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 1 Mathematical Appendix1 Mathematical Appendix1.1 Chapter A1A1.7 Graph each of the following sets. If the set is convex, give a pro of. If it is notconvex, give a counterexample.Answer(a) (x, y)|y = exThis set is not convex.Any combination of points would be outside the set. For example, (0, 1) and(1, e) (x, y)|y = e, but combination of the two vectors with t =not: (,) /x11e+1222(x, y)|y = e.x(b) (x, y)|y = exThis set is convex.Proof: Let (x, y), (x, y) S = (x, y)|y = e. Since y = eis a continuousxx1122function, it is su cient to show that (tx+ (1 - t)x, ty+ (1 - t)y) S for any1212particular t (0, 1). Set t =. Our task is to show that(x+ x),(y+ y)1111212222S.(y+ y) =(e+ e), since y= efor i = 1, 2. Also,11xxx12112i221+ e) = e= ex 1 ex2xx1(x+x12122222 (ee+ e= 2ex 1 ex2xx1222e- 2ex1 ex2+ e= 0 (e- e)2= 0.xxxx121222(c) (x, y)|y = 2x - x; x > 0, y > 02This set is not convex.For example,,, 1,S = (x, y)|y = 2x - x; x > 0, y > 0. However,119121021021,=,+1,S/1111191221022102(d) (x, y)|xy = 1; x > 0, y > 0This set is convex.Proof: Consider any (x, y), (x, y) S = (x, y )|xy = 1; x > 0, y > 0. For any1122t [0, 1],(tx+ (1 - t)x)(ty+ (1 - t)y) = txy+ t(1 - t)(xy+ xy) + (1 - t)xy22121211122122> t+ (1 - t)+ t(1 - t)(xy+ xy), since xy> 1.221221ii= 1 + 2t- 2t + t(1 - t)(xy+ xy)21221= 1 + 2t(t - 1) + t(1 - t)(xy+ xy)1221= 1 + t(1 - t)(xy+ xy- 2) = 1 i xy+ xy= 0.12211221yyxy+ xy= xy2+ xy1- 2 = y2+ y1- 2 = 0122111y22yyy1212y - 1 - 2yy+ y= 0122(y- y)= 0,2122 1 Mathematical Appendixwhich is always true and therefore, (tx+ (1 - t)x, ty+ (1 - t)y) S which is1212convex.(e) (x, y)|y = ln(x)This set is convex.Proof. Let (x, y) + (x, y) S. Then(y+ y) = (ln(x) + ln(x)).1112212122S is convexif1) + ln(x) = ln(1+ 1)2 ln(x122 x12 x21x) = ln(1+ 1)2 ln(x122 x12 x2(xx)= (1+ 1)1/2122 x12 x2x- 2(xx)+ x= 01/211222x1/2+ x1/2= 012which is always true.A1.40 Sketch a few level sets for the following functions: y = xx, y = x+ xand1212y = min[x, x].12Answerxxx222666@@@@@@@@@-@--xxx111(a) y = xx(b) y = x+ x(c) y = min(x, x)121212Figure 1: Sets to Exercise A1.40A1.42 Let D = [-2, 2] and f : D R be y = 4 - x. Carefully sketch this function.2Using the definition of a concave function, prove that f is concave. Demonstrate thatthe set A is a convex set.Answer Proof of concavity: Derive the first and second order partial derivative:yy2= -2x = -2xx2The first derivative is strictly positive for values x < 0 and negative for values x > 0.The second order partial derivative is always less than zero. Therefore, the function isconcave.Proof of convexity: The area below a concave function forms a convex set (Theorem3 1 Mathematical AppendixA1.13). Alternatively, from the definition of convexity the following inequality shouldhold 4 - (tx+ (1 - t)x)= t(4 - (x)) + (1 - t)(4 - (x)). Multiply out to get12212224 - (tx+ x- tx)= 4 - x+ t[(x)- (x)]. Again, the area below the function forms1222212222a convex set.y6x-Figure 2: Graph to Exercise A1.42A1.46 Consider any linear function f (x) = a x + b for a Rand b R.n(a) Show that every linear function is both concave and convex, though neither is strictlyconcave nor strictly convex.Answer The statement is true i , for any x, xR, t [0, 1], it is true that12nf (tx+ (1 - t)x) = tf (x) + (1 - t)f (x).1212Substituting any linear equation in this statement givesf (tx+(1-t)x) = a[tx+(1-t)x]+b = tax+(1-t)ax+tb+(1-t)b = tf (x)+(1-t)f (x)12121212for all x, xR, t [0, 1].12n(b) Show that every linear function is both quasiconcave and quasiconvex and, for n > 1,neither strictly so. (There is a slight inaccuracy in the book.)Answer As it is shown in (a) that a linear function is concave and convex, it must alsobe quasiconcave and quasiconvex (Theorem A1.19). More formally, the statementis true i , for any x, xR(x= x) and t [0, 1], we have12n12f (tx+ (1 - t)x) = min[f (x), f (x)](quasiconcavity)1212f (tx+ (1 - t)x) = max[f (x), f (x)](quasiconvexity)1212Again by substituting the equation into the definition, we gettf (x) + (1 - t)f (x) = min[f (x), f (x)]1212tf (x) + (1 - t)f (x) = max[f (x), f (x)] t [0, 1]1212A1.47 Let f (x) be a concave (convex) real-valued function. Let g(t) be an increas-ing concave (convex) function of a single variable. Show that the composite function,h(x) = g(f (x)) is a concave (convex) function.Answer The composition with an a ne function preserves concavity (convexity). As-sume that both functions are twice di erentiable. Then the second order partial deriva-tive of the composite function, applying chain rule and product rule, is defined ash (x) = g (f (x)) f (x)+ g (f (x)) f (x)224 1 Mathematical AppendixFor any concave function,f (x) = 0,g(x) = 0, it should holdh(x) = 0. In222the case the two functions are convex:f (x) = 0 andg(x) = 0, it should hold22h(x) = 0.2A1.48 Let f (x, x) = -(x- 5)- (x- 5). Prove that f is quasiconcave.221212Answer Proof: f is concave i H(x) is negative semidefinite and it is strictly concave ifthe Hessian is negative definite.H = -2 00 -2zH(x)z = -2z- 2z< 0, for z = (z, z) = 0T221212Alternatively, we can check the leading principal minors of H: H(x) = -2 < 0 and1H(x) = 4 > 0. The determinants of the Hessian alternate in sign beginning with a2negative value. Therefore, the function is even strictly concave. Since f is concave, it isalso quasiconcave.A1.49 Answer each of the following questions yes or no, and justify your answer.(a) Suppose f (x) is an increasing function of one variable. Is f (x) quasiconcave?Answer Yes, an increasing function of one variable is quasiconcave. Any convexcombination of two points on this function will be at least as large as the smallest ofthe two points. Using the di erential-based approach, f is quasiconcave, if for anyxand x, f (x) = f (x) f (x)/ x(x- x) = 0. This must be true for any0110010increasing function.(b) Suppose f (x) is a decreasing function of one variable. Is f (x) quasiconcave?Answer Yes, a decreasing function of one variable is quasiconcave. Similarly to (a),f is quasiconcave if for any x, xand t [0, 1], it is true that f (tx+ (1 - t)x) =0101min[f (x), f (x)].01(c) Suppose f (x) is a function of one variable and there is a real number b such thatf (x) is decreasing on the interval (- inf, b] and increasing on [b, + inf ). Is f (x)quasiconcave?Answer No, if f is decreasing on (- inf, b] and increasing on [b, + inf) then f (x) isnot quasiconcave.Proof: Let a < b < c, and let t=[0, 1], ta + (1 - t)c = b. Given the naturec-bbbbc-aof f , f (b) < min[f (a), f (c)]. Then f (ta + (1 - t)c) < min[f (a), f (c)], so f is notbbquasiconcave.(d) Suppose f (x) is a function of one variable and there is a real number b such thatf (x) is increasing on the interval (- inf, b] and decreasing on [b, + inf ). Is f (x)quasiconcave?Answer Yes.Proof: Let a < b < c, for x [a, b], f (x) = f (a) and for x [b, c], f (x) = f (c).Hence, for any x [a, c], f (x) = min[f (a), f (c)].5 1 Mathematical Appendix(e) You should now be able to come up with a characterization of quasiconcave func-tions of one variable involving the words increasing and decreasing.Answer Any function of one variable f (x) is quasiconcave if and only if is either con-tinuously increasing, continuously decreasing or first increasing and later decreasing.1.2 Chapter A2A2.1 Di erentiate the following functions. State whether the function is increasing,decreasing, or constant at the point x = 2. Classify each as locally concave, convex, orlinear at the point x = 2.(a) f (x) = 11x- 6x + 8 f= 33x- 6321increasing locally convex(b) f (x) = (3x- x)(6x + 1) f= 54x- 6x - 1221increasing locally convex(c) f (x) = x- 1f= 2x + 32x1x34increasing locally concave(d) f (x) = (x+ 2x)f= (6x + 6)(x+ 2x)23221increasing lo cally convex- 3x+ 132(e) f (x) = [3x/(x+ 1)]f= 18x x321(x+ 1)33increasing locally concave13(f ) f (x) = [(1/x+ 2) - (1/x - 2)]f= 4- 8- 1241xxxx + 4232increasing lo cally convex1(g) f (x) =edt f= -et2x21xdecreasing locally convexA2.2 Find all first-order partial derivatives.(a) f (x, x) = 2x- x- x2212112f= 2 - 2x= 2(1 - x) f= -2x11122(b) f (x, x) = x+ 2x- 4x2212212f= 2xf= 4x- 411226 1 Mathematical Appendix(c) f (x, x) = x- x- 2x3212212f= 3xf= -2(x+ 1)1122(d) f (x, x) = 4x+ 2x- x+ xx- x2212121212f= 4 - 2x+ xf= 2 - 2x+ x112221(e) f (x, x) = x- 6xx+ x33121212f= 3x- 6xf= 3x- 6x22122112(f ) f (x, x) = 3x- xx+ x2121221f= 6x- xf= 1 - x11221(g) g(x, x, x) = ln x- xx- x22123231313g= 2xg= -x12x- xx- xx- xx- x222223231313- 2x23g= -x3x- xx- x222313A2.4 Show that y = xx+ xx+ xxsatisfies the equation222231123y+ y+ y= (x+ x+ x)2.xxx123123The first-order partial derivatives are y / x= 2xx+ x,21123y/ x= x+ 2xx, and y/ x= x+ 2xx. Summing them up gives2222333112y+ y+ y= x+ x+ x+ 2xx+ 2xx+ 2xx= (x+ x+ x)2.222xxx121323123123123A2.5 Find the Hessian matrix and construct the quadratic form, zH(x)z, whenT(a) y = 2x- x- x22112H = -2 00 -2zH(x)z = -2z+ 2 * 0zz- 2zT221212(b) y = x+ 2x- 4x22212H = 2 00 4zH(x)z = 2z+ 2 * 0zz+ 4zT2212127 1 Mathematical Appendix(c) y = x- x+ 2x322120H = 6x10 -2zH(x)z = 6xz- 2zT22112(d) y = 4x+ 2x- x+ xx- x22121212H = -2 11 -2zH(x)z = -2z+ 2zz- 2zT221212(e) y = x- 6xx- x331212-6H = 6x1-6 6x2zH(x)z = 6xz- 12zz+ 6xzT22112212A2.8 Suppose f (x, x) = x+ x.221212(a) Show that f (x, x) is homogeneous of degree 1.12f (tx, tx) = (tx)+ (tx)= t(x+ x) = t x+ x.222222212121212(b) According to Eulers theorem, we should have f (x, x) = ( f / x) x+( f / x) x.121122Verify this.+ x2221 f (x, x) = x1x+ xx= x= x+ x1222121212x+ xx+ xx+ x222222121212A2.9 Suppose f (x, x) = (xx)and g(x, x) = (xx).22312121221(a) f (x, x) is homogeneous. What is its degree?12f (tx, tx) = t(xx)k = 4421212(b) g(x, x) is homogeneous. What is its degree?12g(tx, tx) = t(xx)k = 99231221(c) h(x, x) = f (x, x)g(x, x) is homogeneous. What is its degree?121212h(x, x) = (xx)h(tx, tx) = t(xx)k = 2532525325121212128 1 Mathematical Appendix(d) k(x, x) = g (f (x, x), f (x, x)) is homogeneous. What is its degree?121212k(tx, tx) = t(xx)k = 3636181212(e) Prove that whenever f (x, x) is homogeneous of degree m and g(x, x) is homoge-1212neous of degree n, then k(x, x) = g (f (x, x), f (x, x)) is homogeneous of degree121212mn.k(tx, tx) = [t(f (x, x), f (x, x))]k = mnmn121212A2.18 Let f (x) be a real-valued function defined on R, and consider the matrixn+0 f f1nff f1111nH=*...... .. .. . ..ff fnn1nnThis is a di erent sort of bordered Hessian than we considered in the text. Here, thematrix of second-order partials is bordered by the first-order partials and a zero tocomplete the square matrix. The principal minors of this matrix are the determinants0 ff12D= 0 f1, D=fff, . . . , D= |H|.*2ff311112nfff11122122Arrow & Enthoven (1961) use the sign pattern of these principal minors to establish thefollowing useful results:(i) If f (x) is quasiconcave, these principal minors alternate in sign as follows: D= 0,2D= 0, . . . .3(ii) If for all x = 0, these principal minors (which depend on x) alternate in signbeginning with strictly negative: D< 0, D> 0, . . . , then f (x) is quasiconcave23on the nonnegative orthant. Further, it can be shown that if, for all x 0, wehave this same alternating sign pattern on those principal minors, then f (x) isstrictly quasiconcave on the (strictly) positive orthant.(a) The function f (x, x) = xx+ xis quasiconcave on R. Verify that its principal212121+minors alternate in sign as in (ii).Answer The bordered Hessian is0 x+ 1 x21.H=x+ 1 0 1*2x1 019 1 Mathematical AppendixThe two principal minors are D= -(x+1)< 0 and D= 2xx+2x= 0. Which2223121shows that the function will be quasiconcave and will be strictly quasiconcave forall x, x> 0.12(b) Let f (x, x) = a ln(x+ x) + b, where a > 0. Is this function strictly quasiconcave1212for x 0? It is quasiconcave? How about for x = 0? Justify.Answer The bordered Hessian is0aax+xx+x1212H=a-a-a*.x+x(x+x)(x+x)22121212a-a-ax+x(x+x)(x+x)22121212The two principal minors are D= -()< 0 for x, x> 0 and D= 0.a22123x+xWhich shows that the function can not be strictly quasiconcave. However, it can be12quasiconcave following (i). For x= x= 0 the function is not defined. Therefore,12curvature can not be checked in this point.A2.19 Let f (x, x) = (xx). Is f (x) concave on R? Is it quasiconcave on R?2221212++Answer The bordered Hessian is0 2xx2xx221221.2xx2x4xxH=*22112222xx4xx2x2221211The two principal minors are D= -(2xx)< 0 and D= 16xx= 0. Which244212312shows that the function will be strictly quasiconcave. Strict quasiconcavity impliesquasioncavity.A2.25 Solve the following problems. State the optimised value of the function at thesolution.(a) min= x+ xs.t. xx= 122x,x121212x= 1 and x= 1 or x= -1 and x= -1, optimised value= 21212(b) min= xxs.t. x+ x= 122x,x121212x= 1/2 and x= - 1/2 or x= - 1/2 and x= 1/2, optimised value= -1/21212(c) max= xxs.t. x/a+ x/b= 122222x,x121212x= a/3 and x= 2b/3 or x= - 2b/3, optimised value=2ab22221223/23(d) max= x+ xs.t. x+ x= 144v2x,x121212x=1/2 and x=1/2, optimised value== 23/4434412(e) max= xxxs.t. x+ x+ x= 123x,x,x112323123x= 1/6 and x= 1/3 = 2/6 and x= 1/2 = 3/6, optimised value= 1/432 = 108/6612310 1 Mathematical AppendixA2.26 Graph f (x) = 6 - x- 4x. Find the point where the function achieves its2unconstrained (global) maximum and calculate the value of the function at that point.Compare this to the value it achieves when maximized subject to the nonnegativityconstraint x = 0.Answer This function has a global optimum at x = -2. It is a maximum as the second-order partial derivative is less than zero. Obviously, the global maximum is not a solutionin the presence of a nonnegativity constraint. The constrained maximization problem isL(x, z, ) = 6 - x- 4x + (x - z)2The first order conditions and derived equations are:Lx = -2x - 4 + = 0 Lz = - = 0 L= x - z = 0= x - z z = x x = 0If = 0, then x = -2 would solve the problem. However, it does not satisfy the non-negativity constraint. If = 0, then x = 0. As the function is continuously decreasingfor all values x = 0, it is the only maximizer in this range.y6-xFigure 3: Graph to Exercise A2.2611 2 Consumer Theory2 Consumer Theory2.1 Preferences and Utility1.6 Cite a credible example were the preferences of an ordinary consumer would beunlikely to satisfy the axiom of convexity.Answer : Indi erence curves representing satiated preferences dont satisfy the axiom ofconvexity. That is, reducing consumption would result in a higher utility level. Negativeutility from consumption of bads (too much alcohol, drugs etc.) would rather result inconcave preferences.1.8 Sketch a map of indi erence sets that are parallel, negatively sloped straight lines,with preference increasing northeasterly. We know that preferences such as these satisfyAxioms 1, 2, 3, and 4. Prove the they also satisfy Axiom 5 . Prove that they do notsatisfy Axiom 5.Answer : Definition of convexity (Axiom 5 ): If xx, then tx+ (1 - t)xxfor10100all t [0, 1]. Strict convexity (Axiom 5) requires that, if x= xand xx, then1010tx+ (1 - t)xxfor all t [0, 1].100The map of indi erence sets in the figure below represent perfect substitues. We knowthat those preferences are convex but not stricly convex. Intuitively, all combinationsof two randomly chosen bundles from one indi erence curve will necessarily lie on thesame indi erence curve. Additionally, the marginal rate of substitution does not changeby moving from xto x. To prove the statement more formally, define xas convex01tcombination of bundles xto x: x= tx+ (1 - t)x. Re-writing in terms of single01t01commodities gives us:x= (tx, tx) + ((1 - t)x, (1 - t)x). A little rearrangement and equalising the twot00111212definitions results in the equalitytx+ (1 - t)x= (tx+ (1 - t)x), tx+ (1 - t)x). That is, the consumer is indi erent0101011122with respect to the convex combination and the original bundles, a clear violation ofstrict convexity.12 2 Consumer Theoryx26HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHx0HHHHHHHHHHHHrxtrHHHHHHx1r-x1Figure 4: Indi erence sets to Exercise 1.81.9 Sketch a map of indi erence sets that are parallel right angles that kink on theline x= x. If preference increases northeasterly, these preferences will satisfy Axioms121, 2, 3, and 4. Prove that they also satisfy Axiom 5. Do they also satisfy Axiom 4?Do they satisfy Axiom 5?Answer : Convexity (Axiom 5 ) requires that, if xx, then tx+ (1 - t)xxfor10100all t [0, 1].Take any two vectors x, xsuch that x~ x. Given the nature of these preferences, it0101must be true that min[x, x] = min[x, x]. For any t [0, 1] consider the point tx+(1 -001111212t)x. If we can show that min[tx+ (1 - t)x, tx+ (1 - t)x] = min[x, x= min[x, x],00110011212121212then we shown that these preferences are convex. min[tx+ (1 - t)x, tx+ (1 - t)x] =00111212min[tx, tx] + min[(1 - t)x, +(1 - t)x] = min[x, x] + t[min(x, x) - min(x, x)] =01010101011122221122min[x, x]0122Definition of strict monotonicity (Axiom 4): For all x, xR, if x= x, then01n01+xx, while if xx, then xx.010101The map of indi erence sets in the figure below represents perfect complements. Taketwo points x, xalong one indi erence curve. If xx, preferences increase north-0101easterly, then xx. For any two vectors on the same indi erence curve, that is01x= x, it follows xx. Therefore, the definition of strict monotonicity is satisfied0101for these indi erence sets.Strict convexity (Axiom 5) requires that, if x= xand xx, then tx+(1-t)xx1010100for all t [0, 1].Take any two points along the horizontal or vertical part of an indi erence curve suchas (x, x) and (x, x), where x> x. Any convex combination x= x, tx+ (1 - t)x000101t001121222122lies on the same indi erence curve as xand x. Therefore, it is not possible that10xtx+ (1 - t)x. That is, the consumer is indi erent with respect to the convext01combination and the original bundles, a clear violation of strict convexity.13 2 Consumer Theoryx26x0rx1r-x1Figure 5: Indi erence sets to Exercise 1.91.12 Suppose u(x, x) and v(x, x) are utility functions.1212(a) Prove that if u(x, x) and v(x, x) are both homogeneous of degree r, then s(x, x) =121212u(x, x) + v (x, x) is homogeneous of degree r.1212Answer : Whenever it holds that tu(x, x) = u(tx, tx) and tv(x, x) = v(tx, tx)rr12121212for all r > 0, it must also hold that ts(x, x) = u(tx, tx) + v(tx, tx) =r121212tu(x, x) + tv(x, x).rr1212(b) Prove that if u(x, x) and v(x, x) are quasiconcave, then m(x, x) = u(x, x) +12121212v(x, x) is also quasiconcave.12Answer : Forming a convex combination of the two functions u and v and comparingwith m(x) satisfies the definition of quasiconcavity:tWhen u(x) = min tu(x) + (1 - t)u(x) andt12v(x) = min tv(x) + (1 - t)v(x) sot12m(x) = min u(x) + v(x)ttt= t(u(x) + v(x)) + (1 - t)(u(x) + v(x)) .11222.2 The Consumers Problem1.20 Suppose preferences are represented by the Cobb-Douglas utility function, u(x, x) =12Axx, 0 < a < 1, and A > 0. Assuming an interior solution, solve for the Marshal-1-aa12lian demand functions.Answer : Use either the Lagrangian or the equality of Marginal Rate of Substitution andprice ratio. The Lagrangian is14 2 Consumer TheoryL = Axx+ (y - px- px). The first-order conditions (FOC) are1-aa112212L= aAxx- p= 0a-11-ax1121L= (1 - a)Axx- p= 0a-a2x122Lx+ px= 0= y - p1122By dividing first and second FOC and some rearrangement, we get either x=axpor221(1-a)p1x=(1-a)px. Substituting one of these expressions into the budget constraint, results112ap2in the Marshallian demand functions: x=and x=(1-a)y.ay12pp121.21 Weve noted that u(x) is invariant to positive monotonic transforms. One com-mon transformation is the logarithmic transform, ln(u(x)). Take the logarithmic trans-form of the utility function in 1.20; then, using that as the utility function, derive theMarshallian demand functions and verify that they are identical to those derived in thepreceding exercise (1.20).Answer : Either the Lagrangian is used or the equality of Marginal Rate of Substitutionwith the price ratio. The Lagrangian isL = ln(A) + a ln(x) + (1 - a) ln(x) + (y - px- px). The FOC are121122L= a- p= 01xx11L= (1 - a)- p= 02xx22Lx+ px= 0= y - p1122The Marshallian demand functions are: x=ayand x=(1-a)y. They are exactly12ppidentical to the demand functions derived in the preceding exercise.121.24 Let u(x) represent some consumers monotonic preferences overx R. For each of the functions F (x) that follow, state whether or not f also representsn+the preferences of this consumer. In each case, be sure to justify your answer with eitheran argument or a counterexample.Answer :(a) f (x) = u(x) + (u(x))3Yes, all arguments of the function u are transformed equallyby the third power. Checking the first- and second-order partial derivatives revealsthat, although the second-order partial=+6(u(x))()is not zero, the sign2fuu22xxx22iof the derivatives is always invariant and positive. Thus, f represents a monotoniciitransformation of u.15 2 Consumer Theory(b) f (x) = u(x) - (u(x))2No, function f is decreasing with increasing consumption forany u(x) < (u(x))2. Therefore, it can not represent the preferences of the consumer.It could do so if the minus sign is replaced by a plus sign.(c) f (x) = u(x) +nxYes, the transformation is a linear one, as the first partialii=1is a positive constant, here one, and the second partial of the transforming functionis zero. Checking the partial derivatives proves this statement:f=+ 1 anduxxii=.2f2uxx22ii1.28 An infinitely lived agent owns 1 unit of a commodity that she consumes over herlifetime. The commodity is perfect storable and she will receive no more than she hasnow. Consumption of the commodity in period t is denoted x, and her lifetime utilitytfunction is given by8u(x, x, x, . . .) =ln(x), where 0 < < 1.t012tt=0Calculate her optimal level of consumption in each perio d.Answer : Establish a geometric series to calculate her lifetime utility:u = ln(x) + ln(x) + ln(x) + . . . + ln(x)02t012tAs is less than one, this series approaches a finite value. To find the solution, multiplythe expression by and subtract from the original equation [(1)-(2)].u = ln(x) + ln(x) + ln(x) + . . . + ln(x)123t+1012tu - u = (1 - )u = ln(x) - ln(x)t+10t) - ln(x)t+1u = ln(x0t)1 - = ln(x0Thus, the consumers utility maximising consumption will be constant in every perio d.2.3 Indirect Utility and Expenditure1.30 Show that the indirect utility function in Example 1.2 is a quasi-convex functionof prices and income.Answer : The indirect utility function corresponding to CES preferences is: v(p, y) =y (p+ p)-1/r, where r = /( - 1).rr12There are several ways. First, using the inequality relationship, let p= tp+ (1 - t)p.t01We need to show that the indirect utility function fulfills the inequality-1/r-1/r-1/ry p+ p= max[y p+ p, y p+ p]trtr0r0r1r1r12121216 2 Consumer Theorywhich gives:-1/r-1/r-1/ry t(p+ p) + (1 - t)(p+ p)= max[y p+ p, y p+ p].r0r0rr1r1r0r0r1r1r12121212Second, the bordered Hessian can be derived and their determinants checked. Thedeterminants will be all negative.0 p-ppy -ppy-1/rr-1-1/r-1r-1-1/r-112p0 pp-1/r-1/r-1/rH =-ppy pypp((1 - r) - rpp) (1 + r)pppyr-1r-2r-1r-1-1/r-1-1/r-1/r -1r-1-1/r-211112-ppy p(1 + r)pppy ypp((1 - r) - rpp)r-1-1/r-1-1/rr-1r-1-1/r-2-1/r-1r-2r-121222, where p = (p+ p).rr121.37 Verify that the expenditure function obtained from the CES direct utility functionin Example 1.3 satisfies all the properties given in Theorem 1.7.Answer : The expenditure function for two commodities is e(p, u) = u (p+ p)1/rwhererr12r = /( - 1).1. Zero when u takes on the lowest level of utility in U .The lowest value in U is u((0)) because the utility function is strictly increasing.Consequently, 0(p+ p)= 0.rr1/r122. Continuous on its domain R U .n++This property follows from the Theorem of Maximum. As the CES direct utilityfunction satisfies the axiom of continuity, the derived expenditure function will becontinuous too.3. For all p >> 0, strictly increasing and unbounded above in u.Take the first partial derivative of the expenditure function with respect to utility:e/ u = (p+p). For all strictly positive prices, this expression will be positive.rr1/r12Alternatively, by the Envelope theorem it is shown that the partial derivative ofthe minimum-value function e with respect to u is equal to the partial derivativeof the Lagrangian with respect to u, evaluated at (x,), what equals . Un-**boundness above follows from the functional form of u.4. Increasing in p.Again, take all first partial derivatives with respect to prices: e/ p= up(p+ p)(1/r)-1,r-1rrii12what is, obviously, positive.5. Homogeneous of degree 1 in p.e(tp, u) = u ((tp)+ (tp))1/r= tu (p+ p)1/rrr1rr121217 2 Consumer Theory6. Concave in p.The definition of concavity in prices requires1/r1/rt u p+ p+ (1 - t) u p+ p= e(p, u)0r0r1r1rt1212for p= tp+ (1 - t)p. Plugging in the definition of the price vector into e(p, u)t01tyields the relationship1/r1/rt u p+ p+ (1 - t) u p+ p=0r0r1r1r12121/ru t(p+ p) + (1 - t)(p+ p).0r0r1r1r1212Alternatively, we can check the negative semidefiniteness of the associated Hessianmatrix of all second-order partial derivatives of the expenditure function. A thirdpossibility is to check (product rule!)e(p+ p)(p+ p)2rrr1/r2rrr1/r= u (r - 1)p< 0 by r < 0.i12i12pp(p+ p) - r pp(p+ p)22rr2rr2i112i12Homogeneity of degree one, together with Eulers theorem, implies thate/ pp=22ii0. Hence the diagonal elements of the Hessian matrix must be zero and the matrixwill be negative semidefinite.7. Shephards lemmae/ u = (p+p)what is exactly the definition of a CES-type Hicksian demandrr1/r12function.1.38 Complete the pro of of Theorem 1.9 by showing thatx(p, u) = x (p, e(p, u)).hAnswer : We know that at the solution of the utility maximisation or expenditure min-imisation problem e(p, u) = y and u = v(p, y). Substitute the indirect utility functionv into the Hicksian demand function gives x(p, v(p, y)). As the new function is ahfunction of prices and income only, it is identical to the Marshallian demand function.Furthermore, by replacing income by the expenditure function we get the expressionx (p, e(p, u)).2.4 Properties of Consumer Demand1.40 Prove that Hicksian demands are homogeneous of degree zero in prices.Answer : We know that the expenditure function must be homogeneous of degree onein prices. Because any Hicksian demand function equals, due to Shephards lemma,the first partial derivative of the expenditure function and, additionally, we know thatthe derivatives degree of homogeneity is k-1. The Hicksian demand functions must behomogeneous of degree 1 - 1 = 0 in prices.18 2 Consumer Theory1.43 In a two-good case, show that if one good is inferior, the other good must benormal.Answer : The Engel-aggregation in a two-good case is the product of the income elasticityand the repsective expenditure share s+ s= 1. An inferior go od is characterised1122by a negative income elasticity, thus, one of the two summands will be less than zero.Therefore, to secure this aggregation, the other summand must be positive (even largerone) and the other commo dity must be a normal good (even a luxury item).1.55 What restrictions must the a, f (y), w(p, p), and z(p, p) satisfy if each of thei1212following is to be a legitimate indirect utility function?Answer :(a) v(p, p, p, y ) = f (y)pppThe function f (y) must be continuous, strictly in-aaa123123123creasing and homogeneous of degree 0 - a. Each of the exponents ahas to be lessiithan zero to satisfy v decreasing in prices. Furthermore, negative partial derivatives ofv with respect to each price are required to get positive Marshallian demand functionsby using Roys identity.(b) v(p, p, y) = w(p, p)+z(p, p)/y The functions w and z must be continuous and de-121212creasing in prices. Function z has to be homogeneous of degree one and w homogeneousof degree zero: v(tp, tp, ty) = tw(p, p)+(tz(p, p))/(ty) = t(w(p, p) + z(p, p)/y).0101212121212To satisfy v increasing in income, z must be < 0.1.60 Show that the Slutsky relation can be expressed in elasticity form as=-hijijs, whereis the elasticity of the Hicksian demand for xwith respect to price p,hjiijijand all other terms are as defined in Definition 1.6. Answer : The Slutsky relation isgiven byxxhii= x- xijppy .jjMultiplying the total expression with y/y and pgivesjxxxhip= xp- pjjiipjpjyy y.jjBy assuming that x= xbefore the price change occurs, we can divide all three termshiiby x. The result of this operation isixppxyhij= xj- si==- sihpxpxjyxijjiijjijiiAdditional exercise Relationship between utility maximisation and expenditure min-imisationLets explore the relationship with an example of a concrete utility function. A con-sumers utility function is u = x1/2x1/2. For the derived functions see 11219 2 Consumer TheoryStart from the utility function Minimise expenditures s.t. uand derive the Marshallian demand for xto find the Hicksian demand function1x= y/2px= u (p/p)1/2h11211Plug in the respective demand functions to get theindirect utility function expenditure functionv = y/(4pp)e = u(4pp)1/21/21212Substitute the expenditure function Substituting the indirect utility functioninto the Marshallian demand function into the Hicksian demand functionto derive the Hicksian demand function to derive the Marshallian demand functionx= (u(4pp))/2p= u(p/p)x= (p/p)y/(4pp)= y/2p1/21/2h1/21/2112121211211Invert v and replace y by u Invert e and replace u by vto get the expenditure function to get the indirect utility functionv= u(4pp)e= y(4pp)-11/2-1-1/21212Check Roys identity Check Shephards lemma-v/ p=2y(pp)1 / 2= y/2p=u4p= u(p/p)1/211221 e21v/ y4(p3p)p2(4pp)1 /21 /221121Establish the Slutsky equation=-xuy11p2p2p2(pp)1/ 222112substitute u = v(p, y ) into the substitution e ect=-= 0yyx1p4pp4pp21212Table 1: Relationship between UMP and EMP2.5 Equilibrium and Welfare4.19 A consumer has preferences over the single good x and all other goo ds m repre-sented by the utility function, u(x, m) = ln(x) + m. Let the price of x be p, the price ofm be unity, and let income be y.(a) Derive the Marshallian demands for x and m.Answer The equality of marginal rate of substitution and price ratio gives 1/x = p.Thus, the Marshallian demand for x is x = 1/p. The uncompensated demand for mseparates into two cases depending on the amount of income available:m = 0 when y = 1y - 1 when y > 1.(b) Derive the indirect utility function, v (p, y).Answer Again, depending on the amount of income available there will be twoindirect utility functions:when m = 11v (p, y) = lnpy - 1 - ln p when m > 1.20 2 Consumer Theory(c) Use the Slutsky equation to decompose the e ect of an own-price change on thedemand for x into an income and substitution e ect. Interpret your result brie y.Answer A well-known property of any demand function derived from a quasi-linearutility function is the absence of the income e ect. Which can be easily seen in theapplication of the Slutsky equation:xhp = xp + x xyxh+ 0 1p = - 1pp = xp .2Therefore, the e ect of an own-price change on the demand for x equals the substi-tution e ect.(d) Suppose that the price of x rises from pto p> p. Show that the consumer surplus010area between pand pgives an exact measure of the e ect of the price change on01consumer welfare.Answer The consumer surplus area can be calculated by integrating over the inversedemand function of x:1p1CS =- p).10x dx = ln(pp0Calculating the change in utility induced by a price change gives:v = v(p, y) - v(p, y) = y - 1 - ln p- (y - 1 - ln p) = ln(p- p).1110101010As the two expressions are equal, the consumer surplus area gives an exact measureof the e ect of the price change on consumer welfare in the case of quasi-linearpreferences.(e) Carefully illustrate your findings with a set of two diagrams: one giving the indif-ference curves and budget constraints on top, and the other giving the Marshallianand Hicksian demands below. Be certain that your diagrams re ect all qualitativeinformation on preferences and demands that youve uncovered. Be sure to considerthe two prices pand p, and identify the Hicksian and Marshallian demands.01Answer See Figure 6. Please note, that Hicksian and Marshallian demands areidentical here.21 2 Consumer TheoryFigure 6: Graph to 4.1922 3 Pro ducer Theory3 Producer Theory3.1 Production3.1 The elasticity of average product is defined asAP(x). Show that this isxiixAP(x)iiequal to (x) - 1. Show that average product is increasing, constant, or decreasing asimarginal product exceeds, is equal to, or less than average product.Answer : Applying quotient rule to get the first partial derivative of the average productgives:AP(x)f (x)/ x- f (x)i= xii= MP- AP= MP - APxxxxx2iiiiiMultiply this term with the right part of the definition (x/AP ) gives M P/AP - 1 whatiis exactly (x) - 1.iThe first part of the above definition equals the slope of the average product: (MP -AP )/x. It is straightforward to show that whenever marginal pro duct exceeds theiaverage pro duct the slope has to be positive. The average product reaches a maximumwhen the marginal product equals average product. Finally, whenever MP < APaverage pro duct is sloping downwards.3.3 Prove that when the production function is homogeneous of degree one, it may bewritten as the sum f (x) = MP(x)x, where MP(x) is the marginal product of inputiiii.Answer : The answer to this exercise gives a nice application of Eulers Theorem. Thesum of the partial di erentials of a function multiplied with the level of the respectiveinputs is equal to the function times the degree of homogeneity k. The sum of allmarginal products multiplied with input levels gives the pro duction function times k = 1.3.7 Goldman & Uzawa (1964) have shown that the pro duction function is weaklyseparable with respect to the partition {N, . . . , N} if and only if it can be written in1Sthe formf (x) = g f(x), . . . , f(x) ,1(1)S(S)where g is some function of S variables, and, for each i, f(x) is a function of thei(i)subvector xof inputs from group i alone. They have also shown that the pro duction(i)function will be strongly separable if and only if it is of the formf (x) = G f(x) + + f(x) ,1(1)S(S)where G is a strictly increasing function of one variable, and the same conditions on thesubfunctions and subvectors apply. Verify their results by showing that each is separableas they claim.Answer To show that the first equation is weakly separable with respect to the partitions,we need to show that[f(x)/f(x)]= 0 i, j Nand k /N. Calculate the marginalijSSxk23 3 Pro ducer Theorypro ducts of the first equation for two arbitrary inputs i and j:ffSSf(x) = gf(x) = g.ifxjfxSSijThe marginal rate of technical substitution between these two inputs isfSf(x)ixif(x) =fSjxjThis expression is independent of any other input which is not in the same partition NSand, therefore, the production function is weakly separable.(f/f)ij= 0 for k /NSxkTo show that the second equation is strongly separable we have to perform the same ex-ercise, however, assuming that the three inputs are elements of three di erent partitionsi N, j Nand k /NN. The marginal products of the two inputs i and j are:STSTxxS(S)T(T )f(x) = G ff(x) = G f.ijxxijThe MRTS is:f(x)/ xSii.f(x) = ff/ xTjjIt follows for k /NNST(f/f)ij= 0.xk3.8 A Leontief pro duction function has the form y = min {ax, x} for a > 0 and12 > 0. Carefully sketch the isoquant map for this technology and verify that theelasticity of substitution s = 0, where defined.Answer : Taking the total di erential of the log of the factor ratio gives d ln ( x/ax) =21 /xdx- a/xdx. However, the MRTS is not defined in the kinks as the function is2211discontinuous. Along all other segments of the isoquants the MRTS is zero. Therefore,the elasticity of substitution is only defined when the input ratio remains constant. Inthis case, s = 0.3.9 Calculate s for the Cobb-Douglas production function y = Axx, where A >a120, a > 0 and > 0.Answer : The total di erential of the log of the factor ratio givesd ln(x/x) = /xdx-a/xdx. The total di erential of the marginal rate of technical212211substitution givesxa-1d ln Aax= a/ (dx/x- dx/x)121122A xx-1a1224 3 Pro ducer Theoryx26-x1Figure 7: Isoquant map of Leontief technologyPutting both parts together results indx- a/xdxs = /x2211a/ (dx/x- dx/x) = 111223.14 Let y = (nax)1/, wherea= 1 and 0 = < 1. Verify that s=iiijii=1i1/(1 - ) for all i = j.Answer Apply the definition of the elasticity of substitution.) - ln(x))s= (ln(xjiijln (f(x)/f(x))ijx-x11jixx=ji- 1lnax(ax)1/ - 1iiiiiax- 1(sumax)1 / - 1jiiji-x-x11ijxx=ij- 1x-x11ijxxij= -1- 1 = 11 -3.15 For the generalised CES production function, prove the following claims made inthe text.1/nny =ax, wherea= 1 and 0 = < 1iiii=1i=125 3 Pro ducer Theory(a)nlimy =xaii0i=1Answer : Write the log of the CES production function ln y = 1/ ln ax. At = 0,iithe value of the function is indeterminate. However, using LH`opitals rule we can writexln xiilimln y = a.iax0iiAt = 0 this expression turns into ln y = aln x/ a. Because the denominator isiiidefined to be one, we can write the CES production at this point as y = x, what isaiiexactly the generalised Cobb-Douglas form.(b)limy = min {x, . . . , x}1n-8Answer : Let us assume that a= a. Then the CES production function has the formijy = (x+ x). Let us suppose that x= min( x) and < 0. We want to show1/1i12that x= lim( x)1/. Since all commodities xare required to be nonnegative,1-8iiwe can establish x= x. Thus, x= ( x). On the other hand, x= n * x.1/11iii1Hence ( x)= n* x. Letting -8, we obtain lim( x)= x,1/1/1/1-81iibecause limn* x= x.1/-8113.2 Cost3.19 What restrictions must there be on the parameters of the Cobb-Douglas form inExample 3.4 in order that it be a legitimate cost function?Answer : The parameters A, w, wand y are required to be larger than zero. A cost12function is required to be increasing in input prices. Therefore, the exponents a and must be larger zero. To fulfill the property of homogeneity of degree one in input prices,the exponents have to add up to one. To secure concavity in input prices, the secondorder partials should be less than zero. Thus, each of the exponents can not be largerone.3.24 Calculate the cost function and conditional input demands for the Leontief pro-duction function in Exercise 3.8.Answer This problem is identical to the expenditure function and compensated demandfunctions in the case of perfect complements in consumer theory.Because the production is a min-function, set the inside terms equal to find the optimalrelationship between xand x. In other words, ax= x. For a given level of output1212y , we must have y = ax= x. Rearrange this expression to derive the conditional12input demands:x(w, y) = y(w, y) = y12a x .26 3 Pro ducer TheoryThe cost function is obtained by substituting the two conditional demands into thedefinition of cost:yyc(w, y ) = wx(w, y) + wx(w, y) = w121122a + w .3.27 In Fig. 3.85, the cost functions of firms A and B are graphed against the inputprice wfor fixed values of wand y.12(a) At wage rate w, which firm uses more of input 1? At w? Explain?011Answer : Input demand can be obtained by using Shephards lemma, represented by theslope of the cost function. Therefore, at wfirm B demands more of factor 1 and at01wage rate wfirm A has a higher demand of that input.1(b) Which firms production function has the higher elasticity of substitution? Explain.Answer : The first-order conditions for cost minimisation imply that the marginal rateof technical substitution between input i and j equals the ratio of factor prices w/w. Inijthe two input case, we can re-write the original definition of the elasticity of substitutionas/x)/x)x- x212121s = d ln(x,d ln(f/f) = d ln(xd ln(w/w) = w- w121212where the circum ex denotes percentage change in input levels and input prices, respec-tively. Because w= 0, the denominator reduces to w, which is assumed to be the21same for both firms. In (a) we established that input demand at wis larger for firm B01compared to firm A. It follows that the numerator will be larger for B and, subsequently,firm As production function shows the higher elasticity of substitution at w.013.29 The output elasticity of demand for input xis defined asi(w, y )y(w, y) = xiiyyx(w, y) .i(a) Show that(w, y) = f(y)(w, 1) when the production function is homothetic.iyiyGiven a homothetic production function, the cost function can be written as c(w, y) =f(y)c(w, 1). Shephards lemma states that the first order partial derivative with re-spect to the price of input i gives demand of xand to obtain the elasticity we need toitake take the second-order cross-partial derivative of the cost function with respectto output. However, by Youngs theorem it is known that the order of di erentiationdoes not matter. Therefore, the following partial derivatives should be equal:c(w, y)2i= xwy = mcwy .iiPutting everything together gives:cy2(w, y ) == f(y)(w, 1) y(w, 1).iyy wc wiy xif(y)x(w, 1) = 1f (y)iyiiUnfortunately, this is not the result we should get.27 3 Pro ducer Theory(b) Show that= 1, for i = 1, . . . , n, when the production function has constantiyreturns to scale.Answer For any production function with constant returns to scale, the conditionalinput demand xis linear in output level y (see Theorem 3.4). More formally, theiconditional input demand of a production function homogeneous of degree a > 0can be written as x(w, y) = yx(w, 1). By definition, a constant returns to scale1/aiitechnology requires a pro duction function homogeneous of degree 1. Therefore, theconditional input demand reduces to x(w, y) = yx(w, 1). Calculating the outputiielasticity of demand for input xresults in:i(w, y)y(w, y) = xi(w, 1) yiyyx(w, y) = xiy x(w, 1) = 1.ii3.33 Calculate the cost function and the conditional input demands for the linearpro duction function y =nax.iii=1Answer Because the production function is linear, the inputs can be substituted foranother. The most e cient input (i.e. input with the greatest marginal product/ price)will be used and the other inputs will not be used.yif>aj = i, j {1, . . . , n}ajix(w, y) =awwiiji0 if ajj = i, j 1, . . . , n.wwij3.3 Duality in productionAdditional exercise (Varian (1992) 1.6) For the following cost functions indicatewhich if any of properties of the cost function fails; e.g. homogeneity, concavity, mono-tonicity, or continuity. Where possible derive a pro duction function.(a) c(w, y) = y(ww)1/23/412Homogeneity: c(tw, y) = y(twtw)= ty(ww)The function is not1/23/43/21/23/41212homogeneous of degree one.Monotonicity:c(w, y)-1/43/43/4-1/4= 3/4yww> 0 c(w, y)= 3/4yww> 01/21/2ww121212The function is monotonically increasing in input prices.28 3 Pro ducer TheoryConcavity:yw-5/4w3/4yw-1/4w-1/4391/21/2H = -12121616yw1/4w-1/4-yw3/4w-5/491/2-31/221211616|H| < 01y|H| = - 72< 0vw2256w12The function is not concave in input prices.Continuity: Yes1/21/2(b) c(w, y) = vy(2ww)12The function satisfies all properties. The underlying technology is represented byy = xx.12(c) c(w, y) = y (w+ vww+ w)1122The function satisfies all properties. The underlying technology is represented byy = 2/3 (x+ x) + x- xx+ x.22121212(d) c(w, y) = y (we+ w)-w112The function is not homogeneous of degree one. Using Eulers Theorem we getthe resultw= y(we+ we+ w) what is clearly not equal to thec-w2-w11i121wioriginal cost function. Alternatively, it becomes clear from the expression c(tw, y) =ty (we+ w). The function is not monotonically increasing in input prices as-tw112the first partial derivative with respect to wis only positive for prices less than one:1c/ w= ye(1 -w). Furthermore the function is only concave for prices w< 2,-w1111what can be seen from the first determinant of the Hessian matrix: |H| = y(w-112)e.-w1(e) c(w, y) = y (w- vww+ w)1122Monotonicity of the cost function holds only for a narrow set of input prices withthe characteristics 1/4w< w< 4. The conclusion can be derived from the first214partial derivatives and a combination of the two inequalities.cww= y 1 - 12positive for 1 > 12w2w2w111cww112= y 1 - 1positive for 1 > 1or 2 > ww2w2ww2221The function is not concave as the first partial derivatives with respect to bothinput prices are negative and the second-order partial derivatives are positive. Thedeterminants of the Hessian matrix are |H| > 0 and |H| = 0. Thus, the function12is convex.29 3 Pro ducer Theory(f ) c(w, y) = (y + 1/y)vww12The function satisfies all properties, except continuity in y = 0.3.40 We have seen that every Cobb-Douglas pro duction function,y = Axx, gives rise to a Cobb-Douglas cost function,1-aa12c(w, y) = yAww, and every CES production function, y = A (x+ x)1/, gives rise1-aa1212to a CES cost function, c(w, y) = yA (x+ x)1/r. For each pair of functions, show thatrr12the converse is also true. That is, starting with the respective cost functions, workbackward to the underlying production function and show that it is of the indicatedform. Justify your approach.Answer : Using Shephards lemma we can derive the conditional input demand func-tions. The first step to solve this exercise for a Cobb-Douglas cost function is to deriveShephards lemma and to rearrange all input demands in such a way to isolate the ratioof input prices on one side, i.e. left-hand side of the expression. On the right-hand sidewe have the quantity of input(s) and output. Second, equalise the two expressions andsolve for y. The result will be the corresponding production function.1-aw1/(1-a)221x= c(w, y)= ayA w= x1wwwAay111-aw-1/ax= c(w, y)= (1 - a)yA w22= x22wwwA(1 - a)y211(Aay)a1-a= x2x(A(1 - a)y)a1-a1x1-aay = x= axxwhere a = (Aa(1 - a))12a1-aa1-a-1a(1 - a)12a1-aFor the CES cost function a short-cut is used: Derive the conditional input demandfunctions and substitute them into the production function.x= c(w, y)= yAw(w+ w)1-1r-1rrr1w1121x= c(w, y)= yAw(w+ w)1-1r-1rrr2w21221/w+ w-r-ry = (Ay)= Ay12w+ w-r-r123.4 The competitive firmAdditional exercise (Varian (1992) 1.21) Given the following production functiony = 100x1/2x1/4.12(a) Find c(w, w, y).12Answer : Starting from the equality of MRTS and ratio of factor prices, we get30 3 Pro ducer Theoryw/w= 2x/x. Solving for one of the inputs, substituting back in the pro duction1221function and rearranging, we derive the conditional input demand functions:2w1/34/3x= y2and1100w12w-2/34/3x= y2.2100w1Substituting the two functions in the definition of costs, the resulting cost functionis:4/34/3c(w, w, y) = yw2/3(2w)+ yw1/3w2/321/3-2/3121002100121y4/3= 2+ 2w2/3w1/3.1/3-2/310012(b) Find the e ect of an increase in output on marginal cost, and verify that =marginal cost.Answer : Marginal costs areM C = c/ y =(y/100)w2/3w1/32+ 2. Marginal costs are increasing11/31/3-2/31275with output which is shown by2==(y/100)w2/3w1/32+ 2. From the FOC of the La-MCc12-2/31/3-2/312yy15021 / 23/ 4grangian we can derive that=wx=wx. Substituting the conditional input*12121 / 41 /250x25xdemand functions into one of those expressions gives21= w1(2w/w)(y/100)(2w/w)*4/31/3 1/24/3-2/3 -1/450 (y/100)21211/3= 2w2/3w1/31/350 ( y100 )12When you solve the ratios, this expression will be equal to the marginal cost function.(c) Given p = price of output, find x(w, p), x(w, p) and p(w, p). Use Hotellings12lemma to derive the supply function y(w, p).Answer : By maximising p = py - c(w, y) the first-order condition ispy1/3w2/3w1/3(2+ 2) = 01/3-2/3y =p - 1751001233py =100 752+ 22/31/31/3-2/3ww12The first expression a rms the equality of price and marginal cost as the profitmaximum for any competitive firm. The last expression gives already the profit31 3 Pro ducer Theorymaximising supply function. Furthermore, the two following unconditional demandfunctions emerge as solution of this optimisation problem:4/332w1/342p1/34x= 75pww2= 75-2-112+ 212w2+ 2ww1/3-2/31/3-2/331214/33w2/3p4x= 75pww1= 75.-2-122+ 22w2+ 22ww12221/3-2/31/3-2/32/3212The profit function is3100p42w41/312p = 75- 75p+ w2+ 2ww2+ 2ww2ww1/3-2/321/3-2/332/3222211123p4= 75(100 - 75)2+ 2ww21/3-2/3213p4= 25 75.2+ 2ww21/3-2/321Hotellings lemma confirms the output supply function shown above.(d) Derive the unconditional input demand functions from the conditional input de-mands.Answer One, among several, way is to substitute the conditional input demandsinto the definition of cost to obtain the cost function. Calculating marginal cost andequalising with output price, gives, after re-arrangement, the output supply func-tion. Substitution of the output supply function into the conditional input demandsresults in the unconditional input demand functions. Using the example at hand,and starting from the equality c/ y = p gives:y1/3p = 1+ 2w2/3w1/31/3-2/375 210012y = 752+ 2pww 10031/3-2/3 -33-2-1124x(w, p) = 75pww4-3-112+ 2121/3-2/3(e) Verify that the production function is homothetic.Answer : The cost function is a factor of a function of output and input prices.Similarly, the conditional input demand functions are products of a function of yand input prices. Therefore, the possibility to separate the two functions multiplica-tively and following Theorem 3.4 shows that the production function has to be ahomothetic function.(f ) Show that the profit function is convex.Answer : In order to simply this step, I write the constant part of the profit function32 References3as K = 25Calculating the second-order partial derivatives of the profit752+21/ 3- 2 /3function with respect to all prices gives the following Hessian matrix. Be aware ofthe doubble sign change after each derivative with respect to the input price (checkTheorem 3.8).12Kp2-8Kp3-4Kp3w2ww3ww2w2221112---H =-8Kp3-6Kp4-2Kp4wwwwww3432221112----4Kp3-2Kp4-2Kp4w2w2w3w2w2w3121212The own supply e ect is positive, the own demand e ects are negative and all cross-price e ects are symmmetric. Checking the determinants becomes quite tedious.Intuituively, it should become clear that they all have to be positive.(g) Assume xas a fixed factor in the short run and calculate short-run total cost,2short-run marginal cost, short-run average cost and short-run profit function.Short-run total cost are obtained by re-arranging the production function to getxon the left-hand side and plugging in into the definition of cost c(w, y) =1(y/100)w/x1/2+ wx. The first-partial derivative gives the short-run marginal21222cost function smc =w/x1/2. The short-run average costs are equal to sac =1y1250100w/x1/2+.ywx2212100y2Final remark: Some answers might not be the most elegant ones from a mathemat-ical perspective. Any comment and suggestion, also in case of obscurities, are highlywelcome.ReferencesArrow, K. J. & Enthoven, A. C. (1961), Quasi-concave programming, Econometrica29(4), 779800.Goldman, S. M. & Uzawa, H. (1964), A note on separability in demand analysis,Econometrica 32(3), 387398.Varian, H. R. (1992), Microeconomic Analysis, 3rd edn, W. W. Norton & Company,New York.33