plz how do you carry on this

1. A rm has produced E 3} 2 units of {divisible} output at cost c. The rm can sell all or part of its output now, in period 1, or it can wait to sell it in period 2. However, we assume rst that output depreciates between periods, so that if 332 :h 1 units are stored at time 1, then only rg are available for sale at time 2. a. Initially, suppose the selling prices in the two periods are pi\" and p3, respectively, and are known, and that pg 2:: 2p? Assuming the rm wishes to maximise aggregate prots and there is no discounting, characterize the optimal amount to sell in each period. Discuss the effects of changes in E, p? and p3 on the optimal quantities. b. Next, suppose p'i' is known with certainty but it is only known that pg can take one of two values, p3 + E or p3 E, with probabilities El and {1 I9}, respectively, where I9 is known. Assuming the rm maximizes expected prots, determine the amount it will sell in each period as a function of 6". Again assume there is no discounting. Show that the amount the rm sells in period 2 is increasing in H. c. Characterize when the rm would sell more in period 2 under the conditions of part b than in part a. d. Returning to the case in which the prices in both periods are known, suppose the rm discounts future profits by the factor ,0 MUWsuch that for each i E 5'? p512} E S and EH) >1- Ms} A matching is in the core if it is not blocked by an}r set S. A matching n is stable if it cannot be blocked by an individual or by a pair. 1 a. Show that the set of stable matchings is equal to the set of matchings in the core. b. Consider the preference prole 1111 H12 11.13 \"[114 m1 1,3 3,2 2,1 4,3 1712 1,4 2,3 3,2 4,4 where, [nihw = [1,3] means m1 ranks 1111 rst among 1713 3,1 1,4 2,3 4,2 1171.; 2,2 3,1 1,4 4,1 women and mi ranks m1 third among men, etc. Apply the men {M} proposing deferred acceptance algorithm to the above preference prole and obtain the resulting match. c. Showr directlyr {without resorting to a theorem} that the resulting match in part b is stable. :1. Start with the match you obtain in part b and then suppose that women apply the Top Trading Cycle algorithm to trade men, ignoring the preferences of men. What would be the outcome? Describe and explain the outcome. 3. A government needs to procure 10 units of a product. There are two rms it can procure the product from. Each rm independently draws constant marginal cost 1 with probability % and constant marginal cost 2 with probability %. There is no xed cost. These draw probabilities are common knowledge. The rms learn their marginal costs but the information remains private. The government announces its compensation plan that is contingent on reports of the rms of their marginal costs. The rms report their marginal costs truthfully or not. The payoff of a rm when it is assigned :1 units of procurement quota at procurement price p per unit is pg or}? where c is the rm's true marginal cost. a. If both rms report 2, the govermnent pays 2 per unit and splits the order evenly. If both report 1... government pays :r} 1 g :r E 2 per unit and splits the order evenly. If one rm reports 1 and the other reports 2? the government orders the whole procurement from the lower cost {reporting} rm and pays 3;, 1 i: y E '2 per unit. Find the set of I? 3;: that would induce the rms to report their true marginal costs in a Bayesian Nash equilibrium. b. Find the set of any among those given in part a that would minimize the expected procurement cost of the government. 1What is the minimum expected procurement cost?I c. In part a? nd the set of :11:1 3; that would make the rms1 reporting true marginal costs a dominant strategy equilibrium. d. In part c.J nd .5: and 3: that would minimise the expected govermnent procurement cost. What is the minimum procurement cost in this case":I e. Argue directly {without computation} that the minimum expected expenditure of the government in the case of dominant strategy equilibrium cannot be lower than that in the Bayesian Nash equilibrium case. 4. lConsider an economy with one firmj one consumerj two inputs [capital and labor} and one output? food. The consumer owns the rm and initially owns 2 units of capitalJ 4 units of time for leisure or labor and has a strictly increasing? strictly quasiconcave utility function 1:.ij f}? where E :3 U is leisure time consumed and f 2: U is food consumption. The rm produces 117'ny L] units of food when it uses K 23 U units of capital and L 33 D units of labor time. The production function F is homogeneous of degree 1., with strictly positive rst order partial derivatives F1 and F2 and continuous second order partial derivatives y? it}: = 13.23. satisfying F[2?L) 2} U for L { 2 and Fgg[2,L} :2 D for L 3:: 2. a. Interpret the restrictions on F2? in the last sentence above. Why might a production function estimated for a real rm have these properties? b. Justify the claim that the production function F is not concave. c. Show that F [K1 L] is determined by the function F [21 m] You can do this by deriving a formula for FIE, L} in terms of K? L and the function Fm? -.} d. Use the given information to draw a possible feasible consumption set for this economy [the set of all feasible consumption vectors {1 3\"} for the consumer}. Label the axes and intercepts of the upper boundary of the set and explain why you drew the set the way you did. e. Add a possible indifference curve of the consumer to the graph in part {1 in such a way that no competitive {Walrasian} equilibrium exists in this economy. Justify the claim that no competitive equilibrium exists in this case and explain what accounts for that fact. f. Does the conclusion of the second welfare theorem hold in the economy represented by your graph of part e? Explain how you can tell. h. Draw a second graph of the feasible consumption set you drew in part d. Assume now that the economj,r has a competitive equilibrium and draw? in the graphJ a possible indifference curve through the consumer's equilibrium consumption vector (If? f} 35> I}. Draw the consumer's competitive equilibrium budget line in the space of (E: f} Explain why the equilibrium value of the consumer's food consumption in this equilibrium is greater than the value of the labor time the consumer supplies. i. Suppose that the consumer consumes a positive amount of food in a competitive equilibrium. Justify the claim that the consumer supplies at least 2 units of labor time in equilibrium. j. What properties of the fundamentals of the econom].r in part h account for the existence of competitive equilibrium and differ from the case in part e. where no competitive equilibrium exists?I Explain