Two Types of Stock Consider a variation of our Chapter 8 inf

Two Types of Stock. Consider a variation of our Chapter 8 infinite-period \"stock- pricing\" model. The variation here is that there are two distinct \"types\" of stock (rather than just one) that the representative consumer can buy: \"Dow\" stock and \"S&P;\" stock. Denote by a^DOW_t-1 the representative consumer\'s holdings of Dow stock at the beginning of period t and by a^SP_t-1 the representative consumer\'s holdings of S&P; stock at the beginning of period t. Likewise, let S^DOW_t and S^SP_t denote, respectively, the nominal price of Dow and S&P; stock in period t, and D^DOW_t and D^SP_t denote, respectively, the per-share nominal dividend that Dow and S&P; stock pay in period t. The period-t budget constraint of the representative consumer is thus P_t c_t + S^SP_t a^SP_t + S^DOW_t a^DOW_t = Y_t + (S^SP_t + D^SP_t) a^SP_t-1+ (S^DOW_t + D^DOW_t) a^DOW_t-1, in which all of the other notation is standard: Y_t denotes nominal income (over which the consumer has no control) in period t, c_t is real units of consumption, and P_t is the nominal price of each unit of consumption. Also as usual, the lifetime utility of the consumer starting from period t onwards is u(c_t) + beta u(c_t+1) + beta^2 u(c_t+2) + beta^3 u(c_t+3) + ..., where beta (0,1] is the usual measure of consumer impatience. The sequential Lagrangian for this problem is u(c_t) + beta u(c_t+1) + beta^2 u(c_t+2) + ... + lambda_t [Y_t + S^SP_t a^SP_t-1 + D^SP_t a^SP_t-1 + S^DOW_t a^DOW_t-1 + D^DOW_t a^DOW_t-1 - P_t c_t - S^SP_t a^SP_t - S^DOW_t a^DOW_t] + beta lambda_t+1 [Y_t+1 + S^SP_t+1 a^SP_t + D^SP_t+1 a^SP_t + S^DOW_t+1 a^DOW_t + D^DOW_t+1 a^DOW_t - P_t+1 c_t+1 - S^SP_t+1 a^SP_t+1 - S^DOW_t+1 a^DOW_t+1] +... Based on the sequential Lagrangian presented above, what are the algebraic units of the period-t multiplier lambda_t? Carefully justify your answer.

Solution

The diameter of rod produced in a lathe (X) during mass production of nominal dia 20 mm, is known to follow the following distribution: fX(x) = o when x<20 mm = 15 exp(-15*(x-20)) when x>=20 mm Determine the following: 2.1 probability distribution function of diameter of rod (5 points) (1 Point) 2.2 if rods with diameter larger than 20.2 mm are not acceptable, determine the proportion of rods that are accepted and rejected (10 points) (1 Points) 2.3 Find mean and standard deviation of X (5 points) (2 Points)