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Questions:

1.Suppose that the price of milk is Px = $1 per litre, the price of coffee is Py = $4 per cup, and Sally's income is I = $40. Without deriving the optimal Consumption Basket, show that the basket with x = 16 litres of milk, and y = 6 cups of coffee, is NOT optimal.

2.Derive the Expression for Sally's marginal rate of Substitution.

3.Derive Sally's Demand for co_ee as a function of the Variables Px , Py and

4.Derive Sally's Demand for milk as a function of the variables Px , Py and I. (i.e. Do NOT use the Numerical Values for Px , Py and I, from question 1.) For the purposeof this question you should assume an Interior Optimum.

5.Describe the relationship between Sally's Demand for milk and,

(a) Sally's Income;

(b) The Price of Milk;

(c) The Price of Coffee.

6. Suppose that Px = $1 and I = $40. Find the Equivalent Variation for an increase in the Price of Coffee from Py1 = $4 to Py2 = $5.

 

Answers:

1.Utility function is a method of assigning a number to every possible consumer bundle such that the more preferred bundles get assigned larger numbers and vice versa. Sally’s utility function is given by the equation

U(x,y) = xy+2x.

Marginal utility is the rate of change of utility brought about by a small change in the amount of the good being consumed by the individual (Varian, 2010). Here the marginal utility of the good x (milk) is given by the equation

MUx = y+2.

The marginal utility of the good y (coffee) is given by

MUy = x.

To have an optimal solution it must be such that the slope of the indifference curve must be tangent to the price line. Only in that case would there be no other position where a consumer might have been better off. That would imply that the value of the slope of the utility curve and the absolute value of the slope of the price line must be the same. The slope of the utility curve can be found out by

-(MUx / MUy)= -(y+2)/x

Putting the values of x and y given in the question, we get

-(MUx / MUy)= -0.5

The slope of the price line is –(Px/Py). By the information given in the problem,

-(Px/Py)= -0.25

Thus as the two values do not match, we can say that the consumption bundle with x=16 and y=6 is not the optimum bundle.

2.Marginal rate of substitution (MRS) is the maximum amount of good that one consumer is willing to forego so that he or she can obtain an additional unit of another good.  MRS is given by (MUx / MUy). Thus Sally’s marginal rate of substitution is given by

(MUx / MUy)= (y+2)/x.

Again at optimum this must be equal to the absolute slope of the price line which is given by (Px/Py)= 0.25.

Thus Sally’s MRS = (MUx / MUy) = (y+2)/x = (Px/Py) = 0.25.

3.The budget line of Sally can be written as

Pxx+Pyy= I ………… (1)

where I is the total income of the consumer.

Again as per the optimality conditions since it is an interior solution, the consumption bundle will only be optimal when

MRS =(MUx / MUy) = (y+2)/x = (Px/Py)  ………… (2)

It can then be written that

    ((y+2)/x) = (Px/Py)

Or, Pxx = Py(y+2)

Or, Pxx = Pyy+2Py……………. (3)

Or, Pyy = Pxx-2Py………….… (4)

Putting (3) in (1), we get

     Pyy+2Py+ Pyy = I

Or, 2Pyy+ 2Py = I

Or, 2Pyy = I - 2Py

Or, y = (I - 2Py)/2Py

Or, y = (I /2Py)-1

This is the demand curve of coffee (y).

4.We similarly try to find the demand curve for milk (x).

Putting (4) in (1), we get that

      Pxx + (Pxx-2Py) = I

Or, 2Pxx -2Py = I

Or, 2Pxx = I+2Py

Or, x = (I+2Py)/ 2Px …………….. (5)

This is the demand curve for milk.

5.From the equation (5) we might be able to draw some conclusions about the relationship between the demand for milk (x) by Sally and the income, price of milk and price of coffee.

a.From equation (5), we can see that with the increase of income (I), all other variables remaining constant, and the demand for x also rises. There is a direct relationship between the two. Thus for Sally, milk is a normal good.

b.Again from equation (5), we see that with the increase in Px (the price of the milk), the amount of milk demanded falls. Thus there is an inverse relationship between demand for milk and price of milk.

c.In equation (5), with the increase of Py (price of coffee), quantity demanded of milk increases. Thus there is a direct relationship between the price of the other good and the demand of the good. This would suggest that to Sally, milk and coffee are substitutes.

6.When the price of a commodity changes there are two changes that actually happen in obtaining a new optimal. They are that the purchasing power of income is altered and the rate at which we substitute one good for anther changes. Change in demand due to the change in the rate of exchange is known as substitution effect while change in demand due to having more purchasing power is called income effect (Pindyck et al., 2013). Equivalent variation is the change in welfare that is associated with the change in prices.

 

To find the answer, first we need to find the optimum x and y at the original prices.

In fig 1, let the initial budget line be given by RQ and the indifference curve be given by U*. The point of tangency is given by E*(x*,y*).

Using equation (5) and putting the value of I=40, Px=1 and Py=4, we get,

     x = (40 + 2(4))/2(1)

Or, x = 48/2

Or, x* = 24 .

Putting this value of x* in equation (1) along with I=40, Px=1 and Py=4, we get,

     (1)(24)+(4)y = 40

Or, y = (40-24)/4

Or, y = 16/4

Or, y* = 4

Thus the initial optimum bundle (x*,y*) is given by (24,4).

The utility of this consumption bundle is given by U*.

U* = (24)(4)+ 2(24)

Or, U* = 144

Now the price of y has increased to 5 and all the other values remain the same.

The budget line changes to MR and the utility curve is U**. To find the final consumption basket E** which gives (x**, y**), we replace the values in equation (5),

Then,

     x** = (40+2(5))/2

or, x** = 25

Putting these values in equation (1), we get,

      (1)(25)+(5)y** = 40

Or, 5y** = 40-25

Or, y** = 15/5

Or, y** = 3

Thus the final consumption basket is (25,3).

The utility of the consumer U** is given by

U** = (25)(3)+2(25)

Or, U** = 125

Now to find the tangency condition at the decomposition consumption basket A (xa, ya), we have to use the tangency condition. So, we try and find an optimum with the original set of prices and the budget line at TS and the new utility curve U**.

(MUx / MUy) = (y+2)/x = (Px/Py) = 1/4

Then, xa =  4ya+8……(6)

Also, since it is on the same indifference curve i.e. utility level as the final solution,

(xa)( ya) + 2(xa) = 125……… (7)

Using (6) and (7), we might can solve for (xa, ya) using these two as simultaneous equations.

Or, xa2=125.4

Or, xa = 22.36

We get ya=3.59

The cost of this basket is

(22.36)(1) + (3.59)(4)

= 36.72

Thus the equivalent variation is

36.72 – 40

= -3.28.

 

References:

Pindyck, R. and Rubinfeld, D. (2013). Microeconomics. Upper Saddle River, N.J.: Pearson.

Varian, H. (2010). Intermediate microeconomics. New York: W.W. Norton & Co.

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