Sociology 376 Exam 1 Spring 2011 Prof Montgomery

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1 Sociology 76 Exam Spring Prof Montgomery Anwer all quetion. 6 point poible. You may be time-contrained, o pleae allocate your time carefully. [HINT: Somewhere on thi exam, it may be ueful to know that A = a c b d implie A - = d ad bc c b a.] ) [5 point] Conider a et of four individual {,,, 4} with the influence matrix W = where W(i,j) denote the degree to which i i influenced by j. Further uppoe that the initial vector of opinion i given by x = and that the dynamic of opinion formation are governed by the equation x t = W x t-. a) What opinion will individual hold at time and time? b) Partition the individual into communication clae, and then draw the reduced graph of the influenced by relation on the et of communication clae. [NOTE: You can ue matrix algebra to determine the communication clae and compute the image matrix, but it may be fater to imply determine the reduced graph by inpection.] Which communication clae are open? Which clae are cloed? c) Uing your anwer to part (b), will all opinion converge in the long run? Briefly explain why or why not. What property of the W matrix led to thi reult?

2 ) [8 point] Suppoe individual prefer religion A, religion B, or no religion. (Retated in Markov chain terminology, an individual can be in tate A, B, or N). Further uppoe that intergenerational mobility between religion i characterized by the tranition matrix P = where the row and column are labeled (A, B, N), and P(i,j) indicate the probability that a parent in tate i ha a child in tate j. Initially uppoe the reproduction rate (i.e., the average number of children per individual) i equal to one for all individual regardle of religiou preference. a) Draw the tranition diagram for the zero pattern of the P matrix. I P irreducible? I P primitive? Explain how you can determine thee anwer olely by inpection of the zero-pattern tranition diagram. What qualitative implication would primitivity of P have for the long-run probability ditribution of religiou preference (for an individual great-great-great- - grandchildren)? b) Now uppoe the tranition matrix i given by P = Interpret the econd row of the P matrix. How would you now claify tate B? Given an individual with religiou preference A, compute the expected number of generation before hi decendant prefer religion B. c) Given the tranition matrix from part (b), now uppoe that individual with religiou preference A have (on average) children, while individual with preference B or N have (on average) only child. Uing the row vector x t to characterize the number of individual with each religiou preference in generation t, give the general equation for x t a a function of x. Further auming x = [ ], find x and x. d) In part (c), will all individual prefer religion B in the long run? Will the population continue to grow? Give ome intuition. [HINT: Don't be too quick to apply reult you learned in cla. While we aw a reult of the form "p implie q" where p wa the propoition "RP i a primitive matrix," you cannot logically jump to the concluion that "not p implie not q."]

3 ) [8 point] Two people chooe each period whether to drive on the left-hand ide or righthand ide of the road. We can view thi proce a a Markov chain with tate: () both drive on the left () one drive on the left, the other drive on the right () both drive on the right a) Suppoe the tranition matrix for the Markov chain i ε ε / / ε ε Find the limiting ditribution over tate. In the long run, what proportion of period are pent in tate? in tate? in tate? [HINT: Your anwer will be function of ε.] b) Now uppoe the tranition matrix i ε ε / / ε ε Again find the limiting ditribution. In the long run, what proportion of period are pent in tate? in tate? in tate? [HINT: Again, your anwer will be function of ε.] c) Briefly dicu the "macro-level" and "micro-level" interpretation of the limiting ditribution in thee example. What terminology doe Peyton Young (Journal of Economic Perpective 998) ue to decribe the macro-level and micro-level behavior of thi model? d) Briefly define the concept of tochatic tability. What et of tate are tochatically table in part (a)? What et of tate are tochatically table in part (b)? Conceptually, why i it ueful to determine the et of tochatically table tate? e) Suppoe we fix ε =.. Attempting to find the limiting ditribution in part (b) through imulation analyi, a tudent run, chain of length, with each chain tarting from a randomly choen tate (i.e., / chance of tate or or ). To etimate the limiting ditribution, the tudent conider the final (period ) tate for each chain, and report the proportion of chain ending in each tate. Approximately what reult will the tudent report? I thi actually the limiting ditribution from part (b)? If not, explain why the imulation analyi wa faulty, and how you could improve the imulation analyi.

4 4) [5 point] A population (partitioned into -year age clae) ha the Lelie matrix below. (Recall the demography convention that L(i,j) reflect population flow to age cla i from age cla j.) The eigenvector and eigenvalue of thi matrix are alo reported. >> L % Lelie matrix L = >> [eigvec, eigval] = eig(l) eigvec = i i i i i i i i eigval = i i a) Contruct a numerical life table for thi population. [HINT: The firt column of a life table give probability of urvival (from birth) to each age cla; the econd column give expected number of (-year) period remaining for each cla.] b) Compute the gro reproduction rate (GRR) and net reproduction rate (NRR) for thi population. I the population growing or hrinking? How do you know? c) Find the long-run growth rate, and the long-run probability ditribution of the population over age clae. Doe the population reach thi table growth equilibrium for any initial condition? Briefly explain. 4

5 Sociology 76 Exam Spring SOLUTIONS a) [ pt] x = W x = and x = W x = b) [5 pt] Communication clae are the equivalence clae generated by the compound relation can reach and be reached by. Uing the zero-pattern matrix Z = reachability(z) = (I + Z + Z + Z ) > = reachability(z) & reachability(z)' = The communication clae (determined by the unique row of thi matrix) are {}, {,4}, and {}. The reduced graph i given by {} {,4} {} Clae {} and {} are open, while cla {,4} i cloed. c) [5 pt] Given that cla {,4} i cloed, it i not influenced by outider. Thu, we can olve for the equilibrium opinion vector for that cla in iolation. Cla {} i influenced only by {,4} o will adopt the opinion of {,4}. Cla {} i influenced by {,4} and {} and o will alo adopt thi opinion. Thu, all opinion will converge. Thi reult occur becaue the influence matrix i "centered" there i only one cloed communication cla and it ha a primitive ubmatrix.

6 a) [5 pt] A N B P i irreducible and primitive. Uing the graph, P i irreducible becaue every node can reach every other node (directly or indirectly). P i primitive becaue there i a loop. Primitivity of P implie a unique limiting ditribution; every element of the limiting ditribution i poitive; the limiting ditribution doe not depend on initial condition (the individual' initial religiou preference). b) [5 pt] The econd row implie that, once an individual prefer religion B, hi/her children (and grandchildren and great-grandchildren ) will prefer religion B. That i, B i an aborbing tate. Since the tate A and N remain non-aborbing, the Q matrix (characterizing tranition from non-aborbing tate to non-aborbing tate) i given by Q =.8.5. N = I + Q + Q + = (I-Q) - =..5.. =...5. = 5 The um of the firt row of N (= + = ) i the expected number of generation before aborption given initial tate A. c) [5 pt] The dynamic are given by the equation x t = x (RP) t.4. 6 where R = and thu RP = x = [ ] * RP = [.9.5.6] x = [.9.5.6] * RP = [ ] d) [5 pt] Although B i an aborbing tate, individual in tate A have more than enough children to replace themelve even allowing for intergenerational mobility. Note that RP(A,A) =.4. Thi mean that each individual in A average.4 children in A. The number of individual in tate A and the population overall will grow indefinitely. [If you had acce to Matlab, you can how that the population will converge to a table growth equilibrium with growth rate λ =.59 and limiting ditribution [ ].]

7 a) [5 pt] To find the limiting ditribution, we can olve the following ytem: x() = (-ε) x() + (/) x() x() = ε x() + ε [ x() x()] which (after ome algebra) yield x() = / [(+ε)] x() = ε / [(+ε)] = ε / (+ε) x() = x() x() = / [(+ε)] b) [5 pt] The ytem of equation i now x() = ( ε ) x() + (/) x() x() = ε x() + ε [ x() x()] which (after ome algebra) yield x() = / [ + ε + ε ] x() = ε / [ + ε + ε ] x() = ε / [ + ε + ε ] c) [ pt] At the micro level (one pair of individual), the ytem would remain in one convention (LL or RR) for a long time, but occaionally "flip" to the other convention. In Young' terminology, we would oberve "local conformity" with "punctuated equilibria." At the macro level (acro many pair, auming each pair i independent of every other), we would oberve a table ditribution acro convention, with ome proportion of pair in the LL convention and the remainder in the RR convention. In Young' terminology, there i "global diverity." d) [ pt] State i i tochatically table when, a ε become mall, thi tate ha poitive probability in the limiting ditribution (i.e., x(i) > ). A ε, the limiting ditribution in part (a) become [/ /]. Thu, both tate and are tochatically table. A ε, the limiting ditribution in part (b) become [ ]. Thu, only tate i tochatically table. Conceptually, when "mitake" are extremely rare, the Markov chain almot never occupie tate that are not tochatically table. Thu, tochatic tability allow u to determine whether a particular convention i empirically plauible. e) [ pt] Given that ε i very mall and that the length of each chain i very hort, each chain which begin in tate or i likely to remain in the initial tate for the entire period. Chain which begin in tate will tranition immediately to either tate or (with 5% chance) and are then likely to remain in thi tate. Thu, the outcome of the imulation exercie will be determined entirely by the ditribution over initial condition. The tudent will report that approximately half of the chain ended in tate and that the other half ended in tate. Note that [/ /] i not the correct limiting ditribution from part (b). Given ε very mall, the tudent hould have run much longer chain to get the correct reult. The problem i not the number of chain (, would eem adequate) but rather the hort chain length.

8 4a) [ pt] The fundamental matrix i given by N = which can be computed with a calculator. Or, if you had acce to Matlab, >> S = L; S(,:) = [ ]; N = eye(5) + S + S^ + S^ + S^4 N = The firt column of the life table i the firt column of N. The element of the econd column of the life table are given by the column um of N. Thu, the life table i >> lifetable = [(:5)' N(:,) um(n)'] lifetable = b) [5 pt] GRR = =.5. NRR = (.8)(.8) + (.8)(.9)(.7) =.44. Given NRR >, the population i growing. c) [5 pt] The larget eigenvalue of L i growth rate, and the aociated eigenvector (normalized to be a probability vector) i the long-run probability ditribution. Uing the Matlab computation provided, the growth factor i.568 (i.e., the growth rate i 5.68%). The limiting ditribution i found by dividing each of the element of the leading eigenvector by the um of the element (= -.78). Thu, the ditribution given by (the tranpoe of) the vector [ ]. For thi example, individual in the oldet age clae don't have children. If the initial ditribution wa compoed entirely of uch individual, the population would die out. Otherwie, the population will reach the table growth equilibrium decribed above. 4

9 Sociology 76 Exam Spring Prof Montgomery Anwer all quetion. 4 point poible. a b [HINT: Somewhere on thi exam, it may be ueful to know that the matrix c d ha eigenvalue λ = (/)(a + d + qrt(a + 4bc ad + d )) λ = (/)(a + d qrt(a + 4bc ad + d )) ] ) [8 point] Thi emeter, we learned to ue phae diagram in the context of nonlinear ytem. But phae diagram can alo be ued when the ytem i linear. In particular, conider one of the imple intergenerational ocial mobility example we tudied early in the coure. Dynamic were given by x t+ = x t M where x i a row vector, x(i) i the proportion of the population in ocial cla i, and M i the probability tranition matrix M = To implify notation, we may adopt the notation x = [x() x() x()] = [p -p-q q] o that p i the hare of the population in cla, -p-q i the hare of the population in cla, and q i the hare of the population in cla. a) Rewrite the ocial mobility model a a -equation ytem, with p t+ a a function of p t and q t, and q t+ a a function of p t and q t. b) Rewrite the pair of equation from part (a) in delta notation, with p a a function of p and q, and q a a function of p and q. c) Solve for the p-nullcline and q-nullcline. Then olve (algebraically) for the equilibrium (p*, q*) determined by the interection of the nullcline. d) Should the phae diagram be plotted on the unit quare (with p and q ) or a triangular implex (with p+q )? Briefly explain why. Then plot the nullcline on the appropriate diagram. [HINT: Your graph doen t need to be perfect, but hould be qualitatively correct and properly labeled.] e) Ue the pair of equation from part (b) to determine the dynamic in each region of the phae diagram. Then add arrow to your phae diagram to indicate thee dynamic. I the equilibrium table? How can you tell from the diagram?

10 ) [8 point] Conider a public chool ditrict where parent have either high income or low income. Each parent ha one child, and mut decide whether to end the child to the public chool or to a private chool. Low-income parent can t afford private chool tuition, and o alway end their children to the public chool. High-income parent will end their children to private chool if the ratio of low-income to high-income children become too high. To be more precie, let H and L denote the number of high-income and low-income children attending the public chool. Each high-income parent i ha tolerance level θ i, and will end her child to the public chool if and only if θ i > L/H. Tolerance level of high-income parent are uniformly ditributed between and 4. Finally, uppoe that the total number of high-income children i equal to. a) Suppoe high-income parent have adaptive expectation. Write the equation decribing the dynamic of public-chool attendance. [HINT: Your equation hould give H t+ a a function of H t and L. Recall that L i fixed. Make ure your equation hold for every value of H t between and.] b) Suppoe L = 5. Plot the cobweb diagram. Solve for the equilibrium (or equilibria) and indicate whether each i table. What range of initial condition are aociated with each table equilibrium (i.e., what are the bain of attraction)? [HINT: Your cobweb diagram doen t need to be perfect, but I am looking for numerical olution for the equilibria.] c) Suppoe that chool ditrict boundarie are redrawn. There are till high-income children but now 8 low-income children. Redo the analyi from part (b), plotting the new cobweb diagram, olving for the equilibrium, and indicating tability. d) If the chool board want at leat ome high-income children to attend the public chool, what i the maximum number of low-income children that can be placed in the ditrict? Uing terminology from dynamical ytem, decribe how the equilibrium (or equilibria) change a L increae from 5 to 8 and beyond.

11 ) [8 point] Conider the following two-player game between police (who can chooe a high or low level of enforcement) and protetor (who can chooe a high or low level of activity). Following the uual convention in game theory, the pair of payoff are tated a (row player payoff, column player payoff). protetor police H L H, - -, L -,, To analyze thi game uing the replicator dynamic, we can aume two population of player (the row population from which police are drawn; the column population from which protetor are drawn). The equation for the replicator dynamic may written a x = diag(x) [Ay x Ay] y = diag(y) [Bx y Bx] where A = B = x = p p y = q q Subtitution of A, B, x, and y into the Δx and Δy equation yield the pair of equation Δp = p (-p) (q-) Δq = q (-q) (-p) [NOTE: For implicity, I have et period length i equal to (otherwie, there would be an additional h term on the right-hand ide of each equation).] a) Interpret the Δx and Δy equation, explaining the functional form. What i the interpretation of p and q? b) Plot the p-nullcline() and q-nullcline() on a phae diagram. Report all teady tate (p*, q*). How do thee compare to the Nah equilibria of the two-player game? Briefly explain why the replicator dynamic may have teady tate that are not Nah equilibria. c) Finih contructing the phae diagram by drawing arrow to indicate the direction of dynamic in each region. [NOTE: Make ure your phae diagram i well labeled.] d) I the interior teady tate table? [NOTE: To receive full credit, you mut upport your anwer in one of the two following way: () chooe an initial condition cloe to the interior equilibrium, and compute a trajectory for at leat 5 ubequent period, or () derive the Jacobian matrix and compute it eigenvalue. Either way, you might firt want to rewrite the Δp and Δq equation in the form p t+ = g (p t, q t ) and q t+ = g (p t, q t ).]

12 Sociology 76 Exam Spring Solution a) [ pt] [p t+ p t+ q t+ q t+ ] = [p t p t q t q t ] implie p t+ = p t (.6) + ( p t q t )(.) =. +. p t. q t q t+ = ( p t q t ) (.) + q t (.) =.. p t b) [ pt] c) [ pt] p t+ p t =..7 p t. q t implie Δp =..7p.q q t+ q t =.. p t q implie Δq =..p q p-nullcline: Δp = implie q = (7/) p q-nullcline: Δq = implie q =.. p The interection of the nullcline i determined by (7/) p =.. p which implie p* = /6 =.44 q* = /6 =.967 d) [5 pt] The phae diagram hould be plotted on the triangular implex. Becaue p and q are population hare, the um p+q cannot exceed. The equation p+q = define the hypotenue of the implex. Along that edge, everyone in the population belong to ocial cla. See below for the phae diagram. e) [5 pt] Δp > implie q < (7/)p Δq > implie q <..p See below for phae diagram. The equilibrium (p*, q*) appear to be table becaue all arrow are pointing into the equilibrium.

13 continued) phae diagram for part (d) and (e) q Δq = p Δp = a) [ pt] Auming L/H i between and 4, the proportion of high-income parent ending their children to public chool i (/4)(4 L/H). [Parent with tolerance between L/H and 4 end their children to public chool; (/4) i the height of the denity function.] Thu, the number of high-income parent ending their children to public chool i ()(/4)(4 L/H). The dynamic are thu given by H t+ = 5 L / H t But note that no parent end their children to public chool when L/H > 4. Thu, retating the dynamic more preciely, H t+ = 5 L / H t for H t L/4 = for H t < L/4 Equivalently, H t+ = max{, 5 L / H t }

14 b) [ pt] cobweb diagram 9 8 actual H L = 5 L = 8 L = expected H The interior equilibria are determined by the equation H = 5 L/H H H + 5 L = H = ( ± qrt[() 4 () (5 L)]) / For L = 5, thi yield H = 85. or H = 4.6 From the diagram, it i apparent that the equilibria at H* = and H* = 85. are table; the equilibrium at H* = 4.6 i untable. H t will converge to H* = 85. for any initial condition H > 4.6; H t will converge to H* = for any initial condition H < 4.6. c) [ pt] See the cobweb diagram above. Given L = 8, the interior equilibria are given by H* = 7.4 and H* = 7.6. H t will converge to H* = 7.4 for any initial condition H > 7.6; H t will converge to H* = for any initial condition H < 7.6. d) [ pt] If the chool board want to retain any high-income children, it cannot allow L >. A hown on the cobweb diagram, an increae in L caue a downward hift in the generator function. If L increae beyond, there i a catatrophe. Intead of three equilibria, there i a unique equilibrium at H* =.

15 a) [5 pt] The replicator dynamic aume that action become more popular when they yield above-average payoff, and become le popular when they yield below-average payoff. The Δx equation reflect the change in the ditribution of action in the police (row) population. From the perpective of the police, the expected payoff to each action i given by Ay (where y i the ditribution of protetor acro action), and the average payoff for all police i given by x Ay (where x i the ditribution of police acro action). Thu, action i i becoming more popular among police when (Ay)(i) x Ay i poitive. The Δy equation, which reflect change in the protetor (column) population, can be interpreted imilarly. Note the p i the proportion of police chooing high enforcement, while q i the proportion of protetor chooing high activity. b) [ pt] Δp = implie p = or p = or q = /. Δq = implie q = or q = or p = /. q Δq= Δp = Δq = / Δp = / p Recognizing that the horizontal axi i a q-nullcline, and that the vertical axi i a p-nullcline, there are 5 teady tate: (p* =, q* = ), (p* =, q* = ), (p* =, q* = ), (p* =, q* = ), and (p* = /, q* = /). The latter i a mixed-trategy Nah equilibrium of the two-player game. The other teady tate are not Nah equilibria. Given the functional form for the replicator dynamic, any action which i initially zeroed out will never enter the population. c) [ pt] From the Δp and Δq equation, we ee that Δp > when q > /, and that Δq > when p < /. Thi implie the arrow hown in the phae diagram above. d) [ pt] The interior teady tate i untable. Given g (p,q) = p + p(-p)(q-) and g (p,q) = q + q(-q)(-p), the Jacobian matrix i given by J = g / p g / q + ( p)(q ) p( p) = g / p g / q q( q) + ( q)( p) / Evaluated at (p* = /, q* = /), J =, which ha eigenvalue equal to ± (/)i / Becaue ab( ± (/)i) =.444 >, the teady tate i untable. 4