Question

Consider the following utility function of 3 goods, x, y and z: U(x,y,z)=ax+by+cz; x,y,z≥0 and a, b, c are constants. The prices of good x and y is denoted by pX and pY respectively. The income is denoted by m. Good z is provided by the government free of cost but the quantity of good z provided by the government depends on the consumption of good x and y chosen by the consumer. For example, if in equilibrium, the consumer chooses to consume 10 units of good x and 20 units of good y, then government will provide max{10,20}=20 units of good z

So z*=max{x*, y*}, where z*, x*, y* denote the consumption of each good that maximizes utility of the consumer. a) Find the consumption bundle that maximizes utility for the consumer. [6 points] b) Suppose now, the government changed it’s rule for provision of free good z, now z*=min{x*, y*}. Will that change your answer to the previous part? If yes, then find out consumption bundle that maximizes utility for the consumer. If no, just write “consumption in equilibrium remains the same”.

Answer 1

a) The consumption bundle=ax+by+cz =10+20+6 units=26 units

so final consumption bundle = 26*20/10=52 units

b) since the goveernment changes the provision of free good S .A consumption bundle is a set of goods that a consumer may choose to consume. Suppose the only goods available in the world are tea and coffee. Then a consumption bundle is any combination of cups of tea and coffee that the person could choose, and you can write.It changes the provision og all the free goods and services and also maximises utility.A decrease in the price of any good in the consumption bundle also leads to an increase in purchasing power. The impact of this change is known as the income effect where, with this increase in purchasing power, the consumer will buy more normal goods and fewer inferior goods.A consumer chooses her bundle based on her taste and preferences. But, there is also a great amount of emphasis placed on the consumer's income and the price of the good. This also derives the utility.Consumption is separated from production, logically, because two different economic agents are involved. In the first case consumption is by the primary individual; in the second case, a producer might make something that he would not consume himself. Therefore, different motivations and abilities are involved. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual buyer on the hypothesis of constrained optimization. Prominent variables used to explain the rate at which the good is purchased (demanded) are the price per unit of that good, prices of related goods, and wealth of the consumer.

c) The equilibrium price is the market price where the quantity of goods supplied is equal to the quantity of goods demanded. This is the point at which the demand and supply curves in the market intersect.

The supply and demand curves intersect at P* and Q*, which are the equilibrium price and quantity.It's one thing to be able to identify the equilibrium price on a graph, but you should also be comfortable figuring out the price algebraically. Here are the supply and demand curve formulas for this example: Qd = 50 - 5P and Qs = 5 + 10P. The supply curve is denoted as Qs, and the demand curve is denoted as Qd. They are both written as a function of price. If you happen to get formulas that are price as a function of quantity