Question

3. There are two consumers in the market, Jim and Donna. Jim’s utility function is U = xy, (with MUx = y and MUy = x) and Donna’s utility is U = x2y (with MUx = 2xy and MUy = x2). Jim’s income is $100 and Donna’s is $150.

a. Find the demand curves for Jim and Donna when PY = $1.

b. On the graph below, draw Jim’s Demand and Donna’s Demand.

c. Compute the Market Demand (Jim + Donna), and graph on the same set of axes.

Answer 1

Given:

The utility function of Jim and Donna

Jim's Utility function: U_J = xy

Donna’s utility is U_D = x²y

Income of Jim (M_J) = 100

Income of Donna (M_D) = 150

1.) Demand curve for Jim and Donna when the price of good y is $1

that is P_y = $1

Therefore, budget constraint of both;

where Px is the price of good x

Py is the price of god y

x and y are the quantity of good x and good y respectively

Jim's budget constraint

Donna's budget constraint;

Demand curve is determined by the utitiy maximisation condition. That is Jim will demand quantity of x where the it maximises his utility as well as satisfies his budget constraint.

Where the MRS is the marginal rate of substitution that is the ratio of marginal utilitte sand ratio of prices must be equal.

Therefore this condition must satisfy.

Jim's demand function:

Marginal utilities of Jim' are given:

Marginal utility of x (MU_x) = y

Marginal utility of y (MU_y) = x

Since the marginal rate of substitution is the ratio of marginal utilities of both the goods.

Therefore,

Setting it equal to the ratio of prices of both the goods.

Price of good y = $1

Putting this into the budget constraint of him,

Therefore this is the Jim's demand function.

Jim's demand function: x = 50/P_x

Now donna's demand function:

Marginal utilities of Jim' are given:

Marginal utility of x (MU_x) = 2xy

Marginal utility of y (MU_y) = x²

Since the marginal rate of substitution is the ratio of marginal utilities of both the goods.

Therefore,

Setting it equal to the ratio of prices of both the goods.

Price of good y = $1

Putting this into the budget constraint of Donna's,

Therefore this is the Donna's demand function.

Donna's demand function: x = 100/P_x

2.) Drawing the demand curve of both:

Jim's demand curve: x = 50/P_x,

Donna's demand curve: x = 100/P_x

3.) The market demand function is the summation of both individuals.

Market demand function: Jims's demand function + donna's demand function

Market demand function (x): 50/P_x +100/P_x

Market demand function (x):: 150/P_x

Therefore, Marekt demand curve: x = 150/P_x

The market demand curve with jim's and donna's demand curves.

Notice that market demand curve is the summation of jim's and Donna's demand curves.