Please have the answers written down on paper
WRITTEN ASSIGNMENT 3 1. (3 points) On a given year, a real estate company sells 20 penthouses for 2.5 million dollars each and 6 luxury homes for 7.5 million dollars each. At the end of year meeting, a savy agent made the following observation: "for each $100,000 decrease in the price of a penthouse and for each $450,000 decrease in the price of a luxury house, we will sell four more penthouses and two more luxury homes each year." Use this information to suggest the listing price for the properties in the upcoming year that will maximize the revenue. 2. (3 points) Economic theory tells us that production output is the result of the amount of labor and physical capital invested. This analysis produced a calculation that is still in use today, largely because of its accuracy. The Cobb-Douglas production function reflects the relationships between its inputs - namely physical capital and labor - and the amount of output produced. It's a means for calculating the impact of changes in the inputs, the relevant efficiencies, and the yields of a production activity. Here's the basic form of the Cobb-Douglas production function: Q ( L, K ) = alb K1-6 where Q(L, K) is the quantity produced from the inputs L and K; L is the amount of labour expended, which is typically expressed in hours; K represents the amount of physical capital input, such as the number of hours for a particular machine, operation, or perhaps factory; a represents the total factor productivity (TFP) that measures the change in output that isn't the result of the inputs. Typically, this change in TFP is the result of an improvement in efficiency or technology; b and 1 - b reflect the output elasticity of the inputs. Output elasticity is the change in the output that results from a change in either labor or physical capital. Show that for the Cobb-Douglas production function restricted to wL + rK = C, (known as the isocost line) where w > 0 and r > 0 are the costs of labour and capital, respectively, and C is the budget available, the maximum possible production is 8 = a ( " ) " ( 1 - " ) c . 3. (4 points) Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, technological innovation, or cultural fad among a population. The members of the population can be divided into two classes: those who have the information and those who do not. Sociologists often assume that the rate at witch information spread is proportional to the number who have the information and the number who do not. Suppose y(t) denotes the number of individuals who have the information within t time units in a population of N people. (a) Find a mathematical model for social diffusion. (b) Find the general solution to the model found in (a)
