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Partial Equilibrium There is a market of 100 identical consumers of milk and a large number of suppliers

Economics

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Partial Equilibrium

There is a market of 100 identical consumers of milk and a large number of suppliers. The market inverse supply curve is P(Q) = 4Q. Each consumer has preferences

 

 

where q is the quantity of milk the agent buys and m is the quantity of money they have left over. Assume each consumer has enough income to buy as much milk as they like. Also: note that this utility function is not completely standard, as each consumer wants at most 10 units of milk. Therefore, constrain your analysis to this case. Assume no consumer ever consumes more than 10 units.

1. Find the market demand for milk. Specify each step of your derivation.
2. Find the market equilibrium for milk. Calculate consumer and producer surplus.
3. Suppose the government imposes a tax of 1 per unit on milk.
   1. Find the new equilibrium.
   2. What is the change in producer surplus induced by this tax?
   3. What is the change in consumer surplus?
4. Suppose the milk suppliers merged into a monopoly.
   1. What would be the monopoly price?
   2. Compare the consumer and producer surplus in this case to the competitive case.

A firm

A firm has cost function 2q² + 10.

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1. Write its profit function, letting p be the market price of what it sells.
2. Find it’s supply function, writing out each step as follows:
   1. Find the minimal price at which it will supply anything at all.
   2. Find the curve that shows how supply will be determined when the price is higher than this minimal price.
3. The above supply function may have been generated by a production function of the form f(K,L) = K^2/6L^1/6 in the presence of fixed costs.
   1. Does this function demonstrated decreasing, constant, or increasing returns to scale.
   2. Suppose the firm has 10 units of capital. The price of capital is 1 and the price of labor is 2. Assume that it cannot, over the short run, change the amount of capital it has. What is the firm’s short run supply curve?

General Equilibrium

Alice and Bob have two divisible commodities denoted typically as x and y. Alice has 4 of x and 4 of y. Bob has 6 of x and 6 of y.

Alice’s preferences can be represented by the function

u_A(x,y) = 3log(x) + log(y)

and Bob’s by

u_B(x,y) = log(x) + 3log(y).

1. Calculate the Pareto set.
2. Find a Walrasian equilibrium (price and quantities).
3. Suppose instead that Alice is endowed with 8 units of x. Repeat 1 and 2. How has the relative price of x changed? How has welfare changed?

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