Online Intermediate Microeconomics

You need to answer all the questions.

Part (I)

There are 10,000 identical individuals in the market for commodity X, each with a demand function given by d (X) = 8 - p, and 1,000 identical producers of commodity X, each with a function given by s (X) = 10p - 20.

(a) Please find the market demand function.

(b) Please find the market supply function.

(c) Please calculate the market equilibrium price and quantity mathematically.

(d) Given p is on the vertical axis, and quantity is on the horizontal axis, please plot the market demand curve and the market supply curve for commodity X and show the equilibrium point on one set of axes.

(e) Please calculate the producer surplus.

(f) Please calculate the consumer surplus.

Part (II)

Suppose all the conditions are the same as in Part (I), now the government decides to grant a subsidy of $1 on each unit of commodity X produced to each of the 1000 identical producers of commodity X.

(a) Please calculate the market equilibrium price and quantity with subsidy.

(b) Please calculate the producer surplus with subsidy.

(c) Please calculate the consumer surplus with subsidy.

(d) Please use your words to briefly explain why both consumer surplus and producer surplus in Part (II) should have increased after producers receive subsidy from the government (when it is compared with the case in Part (I)).

Part (III)

Suppose all the conditions are the same in Part (I), the government decides to collect a sales tax of $1 per unit sold, from each of the 1,000 identical sellers of commodity X.

(a) Please calculate the market equilibrium price and quantity with tax collection.

(b) Please use your words to explain who actually pays the tax. Is the tax partially transferred to consumers or is it all born by producers?

(c) What is the total amount or taxes collected by the government?

(d) Please calculate the producer surplus.

(e) Please calculate the consumer surplus.

(f) Please use your words to briefly explain whether it makes sense to you that both consumer surplus and producer surplus should have decreased with tax collection (when it is compared to the case in Part (I)).

Q2.

Suppose Vivian has a utility function U = X0.4Y0.6, which X and Y are two goods. The prices for X and Y are $4 and $6, respectively. She has $100 in her pocket.

(a) Please explain why Vivian's utility function is a special case of Cobb-Douglas function.

(b) Please write down Vivian's budget constraint to buy both goods.

(c) Please find the optimal quantities of X and Y when Vivian has achieved her maximal utility level given her budget constraint.

(d) Please find the MRS_XY given by MU_X/MU_Y = 2/3

Q3.

Tom, Henry and Dick constitute the entire market for cod. Tom's demand curve is given by:

Q_T = 100 - 2P, for P ≤ 50; Q_T = 0 for P > 50

Henry's demand curve is given by:

Q_H = 120 - 3P for P ≤ 40; Q_H = 0 for P > 40

Dick's demand curve is given by:

Q_D = 140 - 4P for P ≤ 35; Q_D = 0 for P > 35

(a) Please graph Tom's demand curve with vertical axis as price and horizontal axis as quantity.

(b) Please graph Henry's demand curve with vertical axis as price and horizontal axis as quantity.

(c) Please graph Dick's demand curve with vertical axis as price and horizontal axis as quantity.

(d) Please use the individual demand curves to construct the total market demand for cod.

(e) What is Tom's price elasticity of demand at P=25?

(f) What is Henry's price elasticity of demand at P=25?

(g) What is Dick's price elasticity of demand at P=25?

(h) Does it make sense to you that at P=25, the absolute values of these three people's price elasticity of demand follow the order such that Tom's < Henry's < Dick's? Please briefly explain.

Q4. Billie enjoys coffee (C) and tea (T) according to the utility function U (C, T) = C + T.

(a) The price of coffee is $2 and the price of tea is $1. If Billie's income is $4, how much coffee and tea will she buy? Please draw Billie's indifference curves and budget constraint in this situation and show how to use the graph to answer the question.

(b) Now suppose the price of coffee drops to $1. Will Billie be better off?

How much coffee and tea will she buy?

How does the diagram now look like?

(c) What is Billie's marginal rate of substitution? Is Billie's marginal rate of substitution of coffee for tea decreasing?

If not, why do we ordinarily have an assumption for diminishing marginal rate of substitution?