Practice Problems 2 Ozan Eksi) Problem Quadratic Utility Function and Fixed Income) Let s assume that = r; and consumers preferences are represented by a quadratic utility function uc) = c b=2 c 2 : When we used these two assumptions together with the following Euler Equation E t u 0 c t+ ) = + + r u0 c t ) we found that E t c t+ = c t : Using this equality, and assuming that y t = y; a-) Find consumption at time t in terms of wealth and income of the consumers, c t A t ; y), by using the following Present Value Budget Constraint P P + r )i E t c t+i = + r )i E t y t+i + + r)a t and interpret your nding
Solution Using E t c t+ = c t and y t = y together with the PVBC de ned above nds P P + r )i c t = + r )i y + + r)a t since r > 0; it must be true that + r as c t = + r which can be simpli ed to <. Then the above equality can be written y + + r)a t + r c t = y + ra t Interpretation: Since households earn xed income each period and they have quadratic preferences for consumption, they do not save, and consume the same amount in each period, which is yearly labor income plus the yearly rate of return annuity value) on wealth 2
b-) Find the transitory income yt T, and the relation between A t and A t+ by using the Budget Constraint A t+ = + r)a t + y t c t, and interpret your ndings Solution y T t = y D t y P t = y D t c t = y + ra t ) y + ra t ) = 0 The transitory income is the saving, i.e. the di erence between disposable income and the permanent income. If there is no saving, it must be equal to zero Finding the relation between A t and A t+ A t+ = + r)a t + y t c t = + r)a t + y y + ra t ) = A t Because households consume annuity value of their nancial wealth, ra t, the level of nancial wealth in real terms does not change over periods, thus A t+ = A t 3
Problem 2: Quadratic Utility and Expected Permanent Change in Income Let s assume that = r; and consumer preferences are represented by a quadratic utility function uc) = c b=2c 2 : When we use these two assumptions with the Euler Equation: E t u 0 c t+ ) = + + r u0 c t ) we found that E t c t+ = c t : Using this equality, and also assuming that y i = 0 E t y t+i ) = 2y i = ; 2; ::: 4
a-) Find consumption at time t in terms of wealth and income of the consumers, c t A t ; y), by using the following Present Value Budget Constraint P P + r )i E t c t+i = + r )i E t y t+i + + r)a t and interpret your nding Solution: Using E t c t+ = c t and the de ned time path of y t+i together with the PVBC P P + r )i c t = y + + r )i 2y + + r)a t which can be written as P P + r )i c t = + r )i 2y y + + r)a t since r > 0; it must be true that + r i= <. As a result, the above equality can be 5
written as c t = + r which can be simpli ed to c t = 2y 2y + r r + r y + ra t y + + r)a t Interpretation: Since households have quadratic preferences for consumption, they prefer to consume the same amount each period as long as there is no unexpected change in future income. Hence, even though the change in income is expected to take place in the future, the level of consumption is xed over periods, and it is slightly less than 2y It is less because s/he gains 2y on every period other than today. It is slightly less because assuming the real interest rate is small, r= + r) is a small number) 6
b-) Find the transitory income yt T, and the relation between A t and A t+ under this case by using the Budget Constraint A t+ = + r)a t + y t c t and interpret your ndings Solution: The transitory income is the saving, i.e. the di erence between disposable income and the permanent income y T t = y D t y P t = y D t c t = y + ra t ) 2y r + r y + ra t) = y + r + r y = + r y To nd the relation between A t and A t+, we can use the budget constraint A t+ = + r)a t + y t c t = + r)a t + y 2y r + r y + ra t) = A t + r y Interpretation: Since at time t income gain is less than the future incomes, households, which have quadratic preferences and like to smooth their consumption pattern, either borrow today or use some of their nancial wealth to smooth their consumption. As a result we have found dissaving and that consumers nancial wealth has declined over two time periods. 7
Problem 3: Quadratic Utility Function and Unexpected Permanent Change in Income Let s assume that everything is the same with previous question, except that the consumer does not know at time t whether her/his income will change to 2y: So but the reality is E t y t+i ) = y for i = ; 2; ::: y t+i = y i = 0 2y i = ; 2; ::: and nally, once his income is increased to 2y at time t+, he changes his expectations as E t+ y t+i ) = 2y i = 2; ::: 8
a-) Find the di erence between consumptions at time t + and at time t in terms of wealth and income of the consumers, c t+ A t+ ; y) c t A t ; y); by using the following Present Value Budget Constraint Hint: you need to solve this constraint both at time t and t + ) P P + r )i E t c t+i = + r )i E t y t+i + + r)a t and interpret your nding Solution: Just like the st problem of problem session 2, if consumer thinks that s/he is going to earn y; s/he consumes c t = y + ra t when s/he updates her/his expectations, he starts to consume c t+i = 2y + ra t i = 2; ::: 9
b-) Find the transitory income yt T, and the relation between A t and A t+ by using the Budget Constraint at time t, A t+ = + r)a t + y t c t ), and interpret your ndings Solution: The transitory income is the saving, i.e. the di erence between disposable income and the permanent income y T t = y D t y P t = y D t c t = y + ra t ) y + ra t ) = 0 To nd the relation between A t and A t+ we can use the budget constraint A t+ = + r)a t + y t c t = + r)a t + y y + ra t ) = A t Since saving equals to zero, the wealths at time t and t + are the same 0
Problem 4: Quadratic Utility Function and Expected Temporary Change in Income Let s assume that = r; and consumer preferences are represented by a quadratic utility function uc) = c b=2 c 2 : When we used these two assumptions with the following Euler Equation E t u 0 c t+ ) = + + r u0 c t ) we found that E t c t+ = c t : Using this equality, and also assuming that 2y i = 0; ; 3; 4::: E t y t+i ) = y i = 2
a-) Find consumption at time t in terms of wealth and income of the consumers, c t A t ; y t ), by using the following Present Value Budget Constraint P P + r )i E t c t+i = + r )i E t y t+i + + r)a t and interpret your result Solution: Use E t c t+ = c t and the condition for y t+i de ned above together with the PVBC P P + r )i c t = y + r )i 2y + r) + + r)a 2 t since r > 0 and so <. As a result, the above equality can be written as + r y c t = 2y + r) + + r)a 2 t + r + r which can be simpli ed as ry c t = 2y + r) + ra 3 t 2
b-) Find the transitory income yt T, and the relation between A t and A t+ by using the Budget Constraint A t+ = + r)a t + y t c t )), and interpret your ndings Solution: The transitory income is the saving, i.e. the di erence between disposable income and the permanent income y T t = y D t y P t = y D t c t = 2y + ra t ) 2y To nd the relation between A t A t+ = + r)a t + y t c t = + r)a t + 2y 2y ry + r) + ra t) = ry 3 + r) 3 and A t+, we can use the budget constraint ry + r) + ra t) = A 3 t + ry + r) 3 3
Problem 5: Quadratic Utility Function and Unexpected Temporary Change in Income Let s assume that = r; and consumer preferences are represented by a quadratic utility function uc) = c b=2 c 2 : When we used these two assumptions with the following Euler Equation E t u 0 c t+ ) = + + r u0 c t ) we found that E t c t+ = c t : Now we use an individual who assumes that but the reality is E t y t+i ) = 2y for i = 0; ; 2; ::: 2y i = 0; 2; 3::: y t+i = y i = and nally even though his income decreases to y at time t +, he will realize that this is a temporary change and as a result he does not change his expectations about his future income E t+ y t+2+i ) = 2y i = 0; ::: 4
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a-) Find the di erence between consumptions at time t + and at time t in terms of wealth and income of the consumers, c t+ A t+ ; y t+ ) c t A t ; y t ), by using the following Present Value Budget Constraint P P + r )i E t c t+i = + r )i E t y t+i + + r)a t and interpret your nding Hint: you need to solve this constraint both at time t and t + ) Solution: If consumer thinks that s/he is going to earn y; s/he consumes c t = 2y + ra t when s/he updates her/his expectations, he starts to consume P P + r )i c t++i = + r )i 2y y + + r)a t+ c t+ = 2y r + r y + ra t+ 6
c t+i = 2y r + r y + ra t+i i = 2; ::: c t+ A t+ ; y t+ ) c t A t ; y t ) = r + r y b-) Find the transitory income yt T, and the relation between A t and A t+ by using the Budget Constraint A t+ = + r)a t + y t c t, and interpret your ndings Solution: The transitory income is the saving, i.e. the di erence between disposable income and the permanent income y T t = y D t y P t = y D t c t = 2y + ra t ) 2y + ra t ) = 0 To nd the relation between A t and A t+, we can use the budget constraint A t+ = + r)a t + y t c t = + r)a t + 2y 2y + ra t ) = A t 7
Problem 6 Dynamic Programming) Remember the Bellman Equation which is subject to V t [ + r)a t + y t ] = max c t fuc t ) + + E tv t+ [ + r)a t+ + y t+ ]g A t+ = + r)a t + y t c t Now suppose that there is no labor income and all income is derived from tradable wealth. Further suppose that r = and the utility of the consumer is of the functional form: uc) = c t b=2c 2 t. Use the Bellman equation to solve consumption, c, in terms of A Some additional Information: All CRRA, CARA, and quadratic utility functions are the class of HRRA Hyperbolic absolute risk aversion) utility function 8
Merton shows that the value function is of the same functional form as the utility function for the HRRA utility functions if labor income is fully diversi able, or where there is no labor income at all and all income is derived from tradable wealth As a result, we use a following functional form for V ) Then the Bellman equation V t A t ) = aa 2 t + da t + e V t A t ) = max c t fuc t ) + + E tv t+ A t+ ) can be modi ed as there is no uncertainty, so no need for E t ) aa 2 t + da t + e = maxfc t b=2 c 2 t + c t + [aa2 t+ + da t+ + e] that is subject to A t+ = + r)a t c t 9
FOC w.r.t. A t gives 2aA t + d = + r + 2aA t+ + d) Assuming r = and using the Budget Constraint which can be simpli ed to 2aA t + d = 2a[ + r)a t c t ] + d) A t = [ + r)a t c t ] = A t ra t c t then c t = ra t With quadratic utility and no labor income, households consume annuity value of their nancial wealth Note: Taking derivative only with respect to A t but not c t has turned out to be su cient under this case 20