Derive the first-order kuhn-tucker conditions

Reference no: EM131219829

Question 1. (Consumption under Uncertainty) Consider the representative household who is uncertain about its future income stream, {Y_t}^T=1_t=0. At t = 0, the representative household maximizes the following expected lifetime utility

U = E_o∑_t=0^∞β^tu(C_t)

subject to subject to given A_o ≥ 0, and

A_t+1 = A_t(1+ r) + Y_t - C_t, 0 ≤ t ≤ T -1. (*)

(a) Construct the Lagrangian for this problem, and derive the first-order Kuhn-Tucker conditions for the interior solution. Combining them, obtain the Euler equation and interpret it.

(b) Hereafter, assume (i) u(C) = C - 1/2C², (ii) Ρ (the rate of time preference) equals r. Show that consumption obeys a martingale process. Then, show that consumption has the following representation.

C_t+1 = C_t + ε_t+1

where ε_t+1is the forecasting error.

Question 2. (Lucas Model of CAPM)

Consider the Lucas tree model where there is one asset (tree) that pays a stochastic dividend d_t at time t. The representative consumer maximizes the following lifetime expected utility

U = E₀∑_t=0^∞β_tu(C_t)

subject to (P_t + d_t)s_t>c_t + p_ts_t+1+1q_tb_t+1

(a) Construct the Lagrangian for this problem. Assuming the interior solution, obtain the first order conditions for c_t, s_t+1 and b_t+1. Then, derive the Euler equations for the bond holdings (b_t+1) and asset holdings (s_t+1).

(b) Let p_t and x_t genetically denote the price of an asset and the payoff, respectively. Then, the pricing equations from can be rep resented by P_t = E_tm_t.t+1x_t,t+1. CarefUlly discuss how E_tm_t.t+1, E_tx_t.t+1 and Cov_t(m_t,_t+1,x_t,_t+1) will affect p_t and provide intuitive explanations.

Reference no: EM131219829

Questions Cloud

Explain the trade-off that exists in concurrency control : Define an atomic transaction and explain why atomicity is important.

What will be the monthly payment : What will be the monthly payment if you take out a $100,000 fifteen-year mortgage at an interest rate of 1 percent per month? How much of the first payment is interest and how much is amortization?

Parts of the management style of company : Include these thoughts in your answer: What parts of the management style of this company do you agree and disagree with and why?

What works well and what challenges do you experience : How would you describe your cultural (both intercultural and complementary) relationships? What works well and what challenges do you experience?

Derive the first-order kuhn-tucker conditions : Construct the Lagrangian for this problem, and derive the first-order Kuhn-Tucker conditions for the interior solution. Combining them, obtain the Euler equation and interpret it.

What must be done to add a new relationship : Using the results of your answer to Review Question 8.46, explain what must be done to convert this relationship to 1:1. Use the keys and foreign keys from your answer to Review Question 8.46.

Importance of seasonal forecasting in regards : Find an article through the Hunt Library that discusses the importance of seasonal forecasting in regards to the passenger airline industry.

Prepare schedule of total standard manufacturing costs : The Brown Manufacturing Company's costing system has two direct-cost categories: direct materials and direct manufacturing labor. Manufacturing overhead (both variable and fixed) is allocated to products on the basis of standard direct manufacturing ..

Describe the main line of business of the company : Describe the main line of business of the company. Explain in detail the implementation of the 4 Psmarketing mix concept by the company.

User Account

All Pages