Consider a generic decisionmaker with preferences for two goods, U=Ln(c1)+βLn(c2) and constraints c1= w1-s and c2=...
Consider a generic decisionmaker with preferences for two goods, U=Ln(c1)+βLn(c2) and constraints c1= w1-s and c2= w2 + (1+r)s.
A. Solve the agents decision problem for savings, s=Aw1+ Bw2. (Show Work)
B. What is A?
C. What is B?
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Consider a generic decisionmaker with preferences for two goods,
U=Ln(c1)+βLn(c2) and constraints c1= w1-s and c2=...
Consider a consumer with preferences over current and future consumption given by U (c1, c2) =...
Consider a consumer with preferences over current and future consumption given by U (c1, c2) = c1c2 where c1 denotes the amount consumed in period 1 and c2 the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is m1 = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that p1 = p2 = 1 and let r denote the interest rate. 1. Find...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
2. Consider a consumer with preferences over current and future consumption given by U (C1, C2)...
2. Consider a consumer with preferences over current and future consumption given by U (C1, C2) = (c1)1/(c2)1/2 where c1 denotes the amount consumed in period 1 and ch the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is m1 = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that P1 = P2 = 1 and let r denote the interest rate. (a)...
Consider a household living for two periods
Consider a household living for two periods, t = 1, 2. Let ct and yt denote consumption and income in period t. s denotes saving in period 1, r is the real interest rate and β the weight the household places on future utility. The following must be true about the household’s consumption in the two periods:c1 = y1 − sandc2 = (1 + r)s + y2a. Derive the household’s intertemporal budget constraint.b. Assume that the preferences of the household can be represented by a log utility...
2. Consider a consumer with preferences over current and future consumption given by U(C1, C2) = (c1)1/2 (c2)1/2 where...
2. Consider a consumer with preferences over current and future consumption given by U(C1, C2) = (c1)1/2 (c2)1/2 where cı denotes the amount consumed in period 1 and c2 the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is mı = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that p1 = P2 = 1 and let r denote the interest rate. (a)...
Question 1: Changes in the real interest rate and borrowing constraints Consider the problem of an...
Question 1: Changes in the real interest rate and borrowing constraints Consider the problem of an agent that has an endowment of y- 2 apples in period 1 and y2 in period 2, and takes the real interest rate r as given. Preferences are given by: 4 apples U (c1,c2)log (ci) + log (c2) where cı and c2 denote consumption of apples in period 1 and period 2, respectively. a) Solve for c1,c2 and savings when the real interest rate...
Suppose Sansa lives for two periods. Her preferences are represented as follows: u(c1, c2) = (1+0.8VC2...
Suppose Sansa lives for two periods. Her preferences are represented as follows: u(c1, c2) = (1+0.8VC2 where cı is today's consumption level and c2 is tomorrow's consumption level. Suppose Bob's income today is yı = 100 and his income tomorrow is y2 = 190. Interest rate is denoted by r. 1. Write down Sansa's optimization problem including the budget set. 2. Determine Sansa's optimal consumption bundle (Cl*, C2*) as a function of r.
Consider the following 2-period model U(C1,C2) = min{4C1,5C2} Ci + S = Y1-T C2 = Y2...
Consider the following 2-period model U(C1,C2) = min{4C1,5C2} Ci + S = Y1-T C2 = Y2 - T2 + (1+r)S Where C: first period consumption C2: second period consumption S: first period saving Y] = 20 : first period income Ti = 5 : first period lump-sum tax Y2 = 50 : second period income T2 = 10 : second period lump-sum tax r= 0.05 : real interest rate Find the optimal saving, S*
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e...
hint:
H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi C2 = (*)(1+r)s + y2 where ci and c2 are consumption in periods 1 and 2 respectively; yi and Y2 are income in periods 1 and 2 respectively; s is savings; r is the interest rate on savings.y is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that y> 1; so utility is increasing...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
2. Consider a consumer with preferences over current and future consumption given by U (C1, C2) = (c1)1/(c2)1/2 where c1 denotes the amount consumed in period 1 and ch the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is m1 = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that P1 = P2 = 1 and let r denote the interest rate. (a)...
2. Consider a consumer with preferences over current and future consumption given by U(C1, C2) = (c1)1/2 (c2)1/2 where cı denotes the amount consumed in period 1 and c2 the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is mı = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that p1 = P2 = 1 and let r denote the interest rate. (a)...
Question 1: Changes in the real interest rate and borrowing constraints Consider the problem of an agent that has an endowment of y- 2 apples in period 1 and y2 in period 2, and takes the real interest rate r as given. Preferences are given by: 4 apples U (c1,c2)log (ci) + log (c2) where cı and c2 denote consumption of apples in period 1 and period 2, respectively. a) Solve for c1,c2 and savings when the real interest rate...
Suppose Sansa lives for two periods. Her preferences are represented as follows: u(c1, c2) = (1+0.8VC2 where cı is today's consumption level and c2 is tomorrow's consumption level. Suppose Bob's income today is yı = 100 and his income tomorrow is y2 = 190. Interest rate is denoted by r. 1. Write down Sansa's optimization problem including the budget set. 2. Determine Sansa's optimal consumption bundle (Cl*, C2*) as a function of r.
Consider the following 2-period model U(C1,C2) = min{4C1,5C2} Ci + S = Y1-T C2 = Y2 - T2 + (1+r)S Where C: first period consumption C2: second period consumption S: first period saving Y] = 20 : first period income Ti = 5 : first period lump-sum tax Y2 = 50 : second period income T2 = 10 : second period lump-sum tax r= 0.05 : real interest rate Find the optimal saving, S*
hint:
H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi C2 = (*)(1+r)s + y2 where ci and c2 are consumption in periods 1 and 2 respectively; yi and Y2 are income in periods 1 and 2 respectively; s is savings; r is the interest rate on savings.y is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that y> 1; so utility is increasing...
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