Problem Set 1

Problem Set

¹

1. (Fractions) Write a program to find the value of 1 + ¹₂ + ¹₃ +...+₁₀₀₀¹ .

2. (Double sum) Write a program to find the value of 1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) +...+ (1 + 2 + 3 +..+ 100).

3. (Gcd) Write a function f(x,y) that return the greatest common divisor of x and y.

For example, f(10,15) = 5 and f(21,14) = 7.

4. (Prime seek) Write a program to print all primes between 100 to 200.

5. (Binary) For any integer x such that 0< x <1024 , write a function f(x) to convert x to a based-2 number. For example, f(10) = 1010 and f(15) = 1111.

6. (Fibonacci) Define F as the following: F(0) = 1, F(1) = 2, F(x) = F(x-1) + F(x-2) for x∈ {2,4,5, ...}. Write a program to find the value of F(20).

7. (Digit Reversal) For any integer x ∈ [1111, 9999], write a function f that reverses all digits of a number. For example, f(123) = 321 and f(1345)=5431.

8. (Perfect Number) A perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. For example, 6 is a perfect number because 6 = 1+2+3. Write a program to find all the perfect numbers between 1 and 1000.

9. (Solow 1) Consider following Solow’s model with y_t = k_t^θ, k_t+1 = (1−d)k_t+s.y_t k₁ = 1 The term tdenote a time period. The value of each parameter is as follows:

θ = 0.6,d= 0.1,s= 0.05. In periodt= 1000, the saving rate increases permanently to 10 percent. Simulate the economy and show how the change will affect the growth rate in the short run and the long run.

10. (Matching Problem) There is an economy with 5 men: M₁, M₂, ..., M₅ and 6 women:

W₁, W₂, ..., W₆. You are the social planner. Your role is to arrange marriages for your citizens. Polygamy is forbidden in this economy. The output created by a couple (M_i, W_j) is (i+j)². The output of the economy is the sum of the output of all couples. A single cannot create output. How would you arrange marriages to maximize the economy’s output? ( Note: this question is motivated by the seminal paper of the Nobel Laureate Gary Becker on Theory of Marriage. )

11. (More money; no solution) SEND + MORE = MONEY is a classical “crypto- arithmetic” puzzle: the variables S, E, N, D, M, O, R, Y represent different digits between 0 and 9, and the task is to write a program to find values for then such that the following arithmetic operation is correct: SEND + MORE = MONEY.

For example, S=1, E=2, N=3, D=4, M=5, O=6, R=7 and Y=8 are not correct because 1234 + 56716= 56328.

12. (Factorial) Write a factorial function. For example, fac(5) = 5*4*3*2*1 = 120, fac(3) = 3*2*1 = 6.

1The solution of most (but not all) problems is available on website.

13. (153) A mathematician is very surprised to find that 153 = 1³+ 5³ + 3³. Is there any other number between 100 and 999 that has the same property?

14. (Poor ant) An ant is on the top of a pyramid in which all faces are triangles and the length of each edge is 1 centimeter. (See the picture on the board in front of the class room.) The ant starts jogging on theedges of the pyramid. The ant only runs forward not backward. When the ant passes a vertex of the pyramid, it decides randomly which is the next direction to go. On average, how long (in centimeter) does the ant has to run until it return to the top of the pyramid? You don’t have to count the first vertex and the last vertex that the ant visits.

15. (Solow 2) Consider following Solow’s model with stochastic saving rates with y_t = k_t^θ,k_t+1 = (1−d)k_t+s_t.y_t andk₁ = 1. The termt denotes a time period. The value of each parameter is as follows: θ = 0.6,d= 0.1. s_t is uniformly distributed in [0.1, 0.2]. Calculate, numerically, the normalized S.D.² of output and capital stock of the economy in periods 1000 - 2000. Is output or capital stock more volatile?

16. (Dice) Write a program to solve the following problem numerically. On average how many times do you need to role a dice until all the six faces show up?

17. (Lines in a square) LetA, B, C andDare random points on the perimeter of a unit square. Use the Monte-Carlo method to find the probability that lineAB and line CD intersect.

18. (Central Limit Theorem (CLT)) How accurate is the central limit theorem? : Sup- pose you roll a fair dice 50 times. What is the probability that the average of all the numbers you get is more than 3.55?

- Use a Monte-Carlo simulation to find the answer.

- Use the central limit theorem to find the answer. Calculate the difference between the two answers.

19. (Hypothesis Test)x_i is uniformly distributed in [1, 2]; y_i =a.x_i+e_i. e_i is uniformly distributed between [-1, 1]. Suppose a researcher samples 10 pairs of (x_i, y_i). The researcher knows that yi = a.xi +ei but he does not know the true value of a.

He estimates a by using the following estimator: ˆa = ^P

i

y_i/^P

i

x_i. The researcher wants to perform the hypothesis whethera= 10. His test procedure is to reject the hypothesis if |ˆa−10|> k and accept the hypothesis otherwise.

- Suppose the true value of a= 10. Find the variance of the estimator ˆa.

- Supposek = 0.05. Find the type-1 error of his hypothesis test.

20. (Risk) Consider an economic agent with utility function u(m) =m^θ

wherem is his income andθ is a parameter in [0, 2]. Suppose the agent can choose between the following two income schemes:

2Normalized S.D. ofxis S.D. ofxdivided by the mean ofx.

a) m_a is normally distributed with mean = 10 and variance = 1.

b) m_b is normally distributed with mean = 9.95 and variance = 4.

Find the range of θ in which the agent prefers income schemem_b to m_a, using grid search technique with grid size = 0.01. Hint: you need to find the range of θ in which E[u(m_b)]> E[u(m_a)].

21. (Root Finding) Find the solution of the following equation: x^x = 3 for x ∈ [1,2]

using the 3 following techniques: grid search, bracketing, Newton’s method.

22. (Newron2d) Solve the following system of equations: x²+y²+ 1/x= 20, x^y = 1 by Newton’s method. Use (2, 2) to be the initial point.

23. (Utility Maximization) Consider the following consumer’s optimization problem:

maxx,y x^a+y^a subject to x+ 2y= 1 Plot the value of x that solves the optimization problem and a for a ∈ {0.1,0.2,0.3, ...,2}. Explain the intuition behind the graph you plot.

24. (Anticipation and Policy Changes) Consider the following 50-period optimization problem:

cmaxt,kt+1

50

X

t=1

β^tu(c_t) subject to

c_t+i_t+g_t≤f(k_t) k_t+1 ≤(1−d)k_t+i_t

Suppose u(c_t) = ln(c_t), d = 0.1, β = 0.95, f(k_t) = k_t^1/2, k₁ = 0.1. The term g_t is government spending.

– i) Suppose the government spending g_t = 0 forever. Plot the consumption path of the economy under this scenario.

– ii) Unanticipated Policy Changes 1: In the beginning of 1, the government announces that its fiscal policy would be g_t = 0 forever. The government follows the policyg_t= 0 until period 29. However, in the beginning of period 30, the government announces a permanent policy change and switch tog_t = 1.

After the announcement, g_t = 1 forever. Plot the consumption path of the economy under this scenario.

– iii) Anticipated Policy Changes: In the beginning of period 1, the government announces that its fiscal policy would beg_t= 1 ift >= 30 andg_t= 0 otherwise.

Plot the consumption path of the economy under this scenario.³

– iv) Compare the consumption path in 4.1 - 4.4 and explain the similarities and difference.

25. (2-period stochastic model) Consider the following T-period optimization problem:

cmaxt,kt+1

T

X

t=1

β^tE₁[u(c_t)]

3In this economy, the government is benevolent and never liesintentionally.

subject to

c_t+i_t≤f(k_t) kt+1 ≤(1−d)kt+it

Note that the Euler equation for this optimization is

u⁰(ct) = βEt[u⁰(ct+1)(1 +f⁰(kt+1)−d)]. Suppose u(ct) = ln(ct), d= 0.1, β= 0.95, f(kt) =atk_t^1/2, k1 = 0.1.

The terma_t is the productivity shock in periodt realized in the beginning of period t. Find the maximum expected lifetime utility of the economic agent under the following scenario:

i) T = 2, a1 is always 1. prob(a2 = 1) =prob(a2 = 2) = 0.5.

ii) T = 2, a₁ is always 1. a₂ is normally distributed with mean = 5 and variance

=1.

26. (Non-concavity) Consider the following 2-period optimization problem:

cmaxt,kt+1

2

X

t=1

β^tu(ct) subject to

c_t+i_t≤f(k_t) k_t+1 ≤(1−d)k_t+i_t.

Suppose u(c_t) = c³_t, d= 0.1, β= 0.95, f(k_t) =k^0.9_t , k₁ = 0.1.

- Find the optimal values of c₁, c₂.

27. (Preference Shocks: No Solution) Consider the following problem

cmaxt,kt+1

∞

X

t=1

0.9^tu(c_t) subject to

c_t+k_t+1 ≤k^1/2_t

k₁ is given. Supposeu(c_t) =γ_tln(c_t).The preference shocksγ_t follows the following process:

ˆ

γ_t+1 = 0.75ˆγ_t+_t+1

where t is uniformly distributed between [-0.1, 0.1] and the steady state value of γ_t is 1.

i) Solve for the steady state values of k_t, c_t.

ii) Log-linearizing the Euler equation and the resource constraint to get the sys- tem in form E_t[X_t+1] =WX_t, whereX_t = [ˆc_tkˆ_tˆγ_t]⁰

iii) Solve for ˆc_t as a function of the state variables: ˆk_t, ˆγ_t.

iv) Impulse Response and Monte-Carlo Simulation: Plot the impulse response of ˆ

c_t and ˆy_t to _t with size 0.1. Find the variance of ˆc_t and ˆy_t caused by the preference shocksγ_t, wherey_t is the economy’s output.

28. (Fiscal and Productivity Shocks) Consider the following problem

cmaxt,kt+1

∞

X

t=0

β^tu(ct) subject to

c_t+i_t+g_t≤f(k_t) kt+1 ≤(1−d)kt+it

k0 are given. Suppose u(ct) = ln(ct), f(kt) = atk^α_t, gt is the government spending and is set exogenously by the government. a_t and g_t follows the following process:

ˆ

a_t+1 = 0.65ˆa_t+_a,t+1, gˆ_t+1 = 0.65ˆg_t+_g,t+1⁴

a,t and g,t are i.i.d and normally distributed with zero mean and variance = 0.02.

i) Let a^∗ = 1 and g^∗ be the level of government spending in the steady state.

Solve for the steady state value ofk_t, c_t as function of parameter values andg^∗ and a^∗ (the value of government spending in the steady state).

ii) Parameterization: The data from the Thai economy indicates that the depre- ciation rate is 5 percent per year. The share of income paid to capital to the total output is equal to 66 percent. The annual interest rate is 5 percent. The Thai government spending is about 10 percent of output. Find the values of α, β, d, and g^∗ that match the model with the data.

iii) Log-linearizing the Euler equation and the resource constraint to get the sys- tem in the following matrix form:

E_t[x_t+1] =Wx_t wherex_t= [ˆc_tˆk_tˆa_tˆg_t]⁰

iv) Solve for ˆc_t as a function of ˆk_t, ˆa_t and ˆg_t.

vi) Monte-Carlo Simulation: find the variation in yt that caused by these two shocks together.

29. (Small Open Economy) This question is to be done purely analytically. No linearl- ization, no programming required. Consider the following problem:

cmaxt,ft+1

∞

P

t=1

β^tE₁[u(c_t)]

s.t ft+1≤(1 +r)ft+yt−ct, f1 = 0.

where u_t = αc_t− ^γ₂c²_t, f_t is the stock of foreign asset holding. Assuming that α is very large and γ is a very small positive number. yt is economy output which follows an i.i.d. exogenous process: prob(y_t = 0) = prob(y_t = 1) = 1/2. Note that β = 0.9 and the world interest rater = 1/β−1.

4xˆt≡^xt_x^−x∗^∗ andx^∗ is the steady state value ofx. The process ofatandgtcan actually be estimated from the data of the Thai economy.

i) Find the Euler equation and the traversality condition. Interpret the meaning of the traversality condition.

ii) Solve for the ct as a function of yt and ft.

iii) Is the steady state of the economy unique? Explain.

iv) Can you still solve the problem easily ifu_t=ln(c_t)? Why or why not? Discuss.

30. (Fixed Point) Find the fixed point of the following functionf(x) =x²using iteration method. Usex = 0.5 to be the initial point.

31. (Functional Equation 1) Numerically solve the functional equation using iteration:

V(x) =x+ 0.1V(1−x) forx∈[0,1] using 11 grids. Then plot functionV(x). Can you analytically find V(x)?

32. (Functional Equation 2) Numerically solve the functional equation using iteration:

V(x, y) =x∗y+ 0.1V(y, x) for (x, y)∈[0,1]² using 11x11 grids. Then plot function V(x, y). You may need to use a command for 3 dimensional plot.

33. (Simple Dynamic Programming) Consider the following problem

{ct,kmaxt+1}^∞_t=1

∞

X

t=1

0.95^tln(c_t) subject to

k_t+1 ≤0.9k_t+k^0.66_t −c_t

k1 is given. Suppose at period 1, the level of the capital is at 95 percent of its steady state value. Solve the optimization problem using numerical dynamic programming approach. Then plot the value functionV and the time path of consumption share over output of this economy.

34. (Learning Monopoly) Solve the following problem using numerical dynamic pro- gramming approach and plot the time path of q_t.

{qmaxt}^∞_t=0

∞

X

t=1

0.75^t((1−q_t)q_t−c_tq_t) s.t.

c_t+1 = 1

1 + 10q_t, q₀ = 0 Then solve the optimal time path for qt in case that

c_t+1= 1

1 +q_t, q₀ = 0 . How the results in the two cases are different? Why?

35. (Simple Markov) Consider the following problem

{c_t,kmaxt+1}^∞_t=1

∞

X

t=1

0.95^tln(c_t) subject to

k_t+1 ≤0.9k_t+A_tk^0.66_t −c_t

where k₁ is given, A_t ∈ {A¹, A², A³}, A¹ = 0.99, A² = 1.00, A³ = 1.01. The transition matrix is

Π =







0.3 0.4 ? 0.5 0 0.5 0.4 0.4 0.2







where Π_ij = prob(A_t+1 = A^j|A_t = Aⁱ). What is the missing number? Assuming that the economy start from the steady state in which A_t = 1. Use Monte-Carlo simulation to find the correlation of consumption in period t and consumption in period t+ 1.

36. (Tree Cutting) Consider a farmer problem on when to cut a tree. If he cuts a tree with height h and sells it he will get h baht. After he cut a tree, he can plant a new tree with height 0 with no cost. The growth function of the tree is h_t+1 ={1 + 10h_t}^α. The farmer’s dynamic programming problem is

V(h_t) = max

cut,notcut{βV({1 + 10h_t}^α), h+βV(0)}

Consider the case thatβ =α= 0.9 and find the optimal policy rule for cutting the tree using numerical dynamic programming. Then consider the case that β =α= 0.5. In such case, can you find the optimal policy rule analytically.

37. (Dice Gambling) Consider the following gambling game. The gambler tosses a fair dice. After tossing the dice, the gambler has two choices: continues playing the game or stops. If the gambler stops the game, he will get money in baht equal to the number shown on the last dice he tossed. If he continues, he has to pay 0.5 baht, then roll the dice again to get a new number. And so on. Assume that the discount factor is 1 and formulate the problem as a dynamic programming problem.

Write down the Bellman equation. Then use numerical dynamic programming to find the optimal rule for stop playing.

38. (Auction) Consider a first-priced auction game with two bidders: 1 and 2. Bidder i submits a sealed bid b_i. If b_i > b_j, bidder i gets the auctioned good. The payoff of agents i when getting the good isv_i−b_i, where v_i is hisprivate valuation of the good. v_i is uniformly distributed between [0,1]. Find the symmetric Bayes-Nash equilibrium of this game. Hint: 1) You don’t need calculus for this question and 2) if b^∗_i is the bidding function in the Nash equilibrium, then b^∗_i =b^∗_j and

b^∗_i(v_i) = max

x

h(v_i−x)prob(x > b^∗_j(v_j))ⁱ.

39. (Leisure) Consider the following dynamic macroeconomic problem with leisure and work

{cmaxt,lt}

∞

X

t=1

β^t(lnc_t+ ln(l_t))

subject to k_t+1 = (1−d)k_t+^qk_t(1−l_t)−c_t Note that this problem is equivalent to the following problem

V(k) = max

c,n,k^/

{ln(c) + ln(1−n) +βV(k^/)}

subject to k^/ = (1−d)k+^qk(1−l)−c

where c_t is consumption, k_t is capital, l_t is leisure and 1−l_t is working hours. The value of each parameter are the following: β = 0.9 and d= 0.1.

Possible value of working hours are discrete; the possible values oflt are ¹₄,²₄ and ³₄. i) Use numerical dynamic program to solve for V(k). Then plot the graph of V(k). Use 101 grid points and the upper bound of V is ¯k = _{4(1+βd−β)}^β2 2 and the lower bound is ¯k/100. (Note: You don’t need to solve for the steady state of the model.) Then plot the graph of V(k).

ii) Suppose the economy start from k₁ = ¯k/100. Simulate the time path of working hours n_t of this economy. Interpret the graph you get.

40. (Dynamic Cournot) This question is about a dynamic Cournot model with learning externality. The setting is as follows. There are two firms in the market: firms 1 and firms 2. The objective function of firm i is to maximize its lifetime profit:

{qmax1t}^∞_t=0

∞

X

t=1

0.75^t((10−q_1t−q_2t)q_it−c_itq_it) s.t.

c_it+1 = 1

1 +q_1t+q_2t, q_i0 = 0,{q_jt}^∞_t=0 is given

whereq_it is quantity that firmiproduces in periodt. You are asked to find the time path of q₁ in a (Nash) equilibrium of the game. You may restrict attention on a symmetric Markov equilibrium in which in which q_it only depends on q_1t−1+q_2t−1.