## Find market-clearing price and competitive allocation, Marketing Management

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1. Consider an economy in which George and Harriet consume only ale and bread. Georgeís utility function is

UG = aG(bG 1)

where aG and bG are his consumption of ale and bread. Harrietís utility function is

UH = (aH )2 (bH 1)

where aH and bH are her consumption of ale and bread. George is endowed with one unit of ale and one unit of bread. Harriet is endowed with two units of bread but no ale. Using aG as an index, Önd all of the Pareto optimal allocations in which George and Harriet consume positive amounts of both goods.

2. Consider an economy consisting of three people: Frida, George and Harriet. They consume two goods, ale and bread, and their utility functions are

UF = (aF )2 bF

UG = (aG 1)(bG 1)

UH = aH (bH 1)

Frida is endowed with one unit of ale and one unit of bread. George is endowed with two units of bread and no ale. Harriet is endowed with two units of ale and no bread.

(a) Find each agentís best attainable commodity bundle.

(b) Find the market-clearing price.

(c) Find the competitive allocation.

(d) How much of which good does each person sell in the competitive equilibrium? How of much of which good does each person buy in the competitive equilibrium?

3. Alice and Bob survive on hamburgers and salads. Aliceís utility function is

UA = hA(sA 1)

where hA and sA are her consumption of hamburgers and salads respectively. Bobís utility function is

UB = (hB 1)2 sB

where hB and sB are his consumption of hamburgers and salads respectively. Let p be the price of hamburgers measured in salads. Alice is endowed with one salad and two hamburgers. Bob is endowed with six salads and two hamburgers. Find the competitive equilibrium.

4. An economy is endowed with one unit of ale and one unit of bread. There are two people in the economy, George and Harriet. Georgeís utility function is

UG = aG(bG + 1)

where aG and bG are his consumption of ale and bread. Harrietís utility function is

UH = aH (bH + 1)

where aH and bH are her consumption of ale and bread. Show that the allocation

(aG; bG; aH ; bH ) = (1=4; 0; 3=4; 1)

is Pareto optimal.

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