Let P(t) be a function and Q(t) be functions.
dP/dt = aQ
dQ/dt = -bP
(1) Find the equilibrium point(s) and discuss their
stability.
(2) Find the trajectories of P and Q.
(3) Solve the system to find P and Q as functions of t.
Let P(t) be a function and Q(t) be functions. dP/dt = aQ dQ/dt = -bP (1) Find the equilibrium poi...
If p is the price in dollars of computer mice at time, t, then we can think of price as a function of time. Similarly, 1. then number of computer mice demanded by consumers at any time, and the number of computer mice supplied by producers at any time, may also be considered as functions of time as well as functions of price. Both the quantity demanded and the quantity supplied depend not only on the price, but also on...
Consider the economic model below, where P is the price of a single item on the market and Q is the quantity of the item available on the market. Both P and Q are functions of time that can be viewed as two interacting species, and a, b, c, and f are positive constants. dQ dP cQ(fP Q) dt aP dt If a 1, b 15,000, c 1, and f= 10, find the equilibrium points of this system and classify...
Let Qd be the number of units of a commodity demanded by consumers at a given time t and let Qsdenote the number of units of the commodity supplied by producers at a given time t. Let p be the price in dollars of the commodity at time t. Suppose the supply and demand functions for a certain commodity in a competitive market are given, in hundreds of units, by Qs = 30 + p + 5 dp/dt Qd =...
1. The inverse demand function for a good takes the constant elasticity form p(Q) = Qβ , −1 < β < 0, which is a commonly used simple functional form. The good is produced by n identical firms with a cost function c(qi) = cqi . Note that c 0 (qi) = c and c 00(qi) = 0; i.e., there are constant marginal costs. A specific tax of t per unit is imposed on the production of the good. (a)...
For the equation (dp/dt)=(P+2)(P^2-6P+5) find the equilibrium points and make a phase portrait of the differential equation. Classify each equilibrium point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves determined by the graphs of equilibrium solutions.
Consider Market Model: Demand: Supply: Q= a - bP Q=-c+dP (a, b > 0) (c, d > 0) * 1) Discuss in words the meaning of each equation in the model (3 points); 2) Find the equilibrium levels of P* and Q* (3 points); 3) Draw qualitative conclusions about changes in P* and Q* when each of the parameters change. (Qualitative conclusion shows the direction of change.) Explain economic meaning of these changes. (Total 6 points: 3 points for P*;...
dP 7. For the equation = (P+2)(P2 - 6P+5)find the equilibrium points and make a phase dt portrait of the differential equation. Classify each equilibrium point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves determined by the graphs of equilibrium solutions. (6pts)
2. Suppose a population P(t) satisfies the logistic differential equation dP dt = 0.1P 1 − P 2000 P(0) = 100 Find the following: a) P(20) b) When will the population reach 1200? 2. Suppose a population P(t) satisfies the logistic differential equation 2P = 0.1P (1–2000) = 0.1P | P(0) = 100 2000 Find the following: a) P(20) b) When will the population reach 1200?
2. Points = 26. Consider Market Model: Demand: Supply: Q=a-bP Q=-c+dP (a, b>0) (c,d > 0) 1) Discuss in words the meaning of each equation in the model (3 points); 2) Find the equilibrium levels of P* and Q* (3 points); 3) Draw qualitative conclusions about changes in P* and Q* when each of the parameters change. (Qualitative conclusion shows the direction of change.) Explain economic meaning of these changes. (Total 6 points: 3 points for P*; 3 points for...
(16 points) Cournot Duopoly. Market demand is p(Q) = 50 – 4Q, where Q = 4+ 42. Firm 1's cost function is C (91) = 0, and firm 2 has a cost function C2(92) = 1092- The two firms engage in Cournot competition; they simultaneously choose a quantity and the price adjusts so that the market clears. (a) Formally write firm 1's profit maximization problem (b) Find firm l's best response function. (c) Take as given that firm 2's best...