Question 1: Lindahl pricing

A competitive economy has three consumers indexeda= 1,2,3. It produces two goods, a private goodXand a public goodG. Consumers of course choose individual amounts ofXto consume, but are restricted to the same quantity ofG.

The economy has a linear ppf with slope -1, that is,Xcan be transformed intoGat a rate of 1 for 1 (or the MRT is 1). Thus the prices ofXandGare equal. Normalise the price ofXto unity, then this is also the cost of producing each unit ofG.

In order to price the public good, the government has declared that each consumer will have to pay a unique fraction of the cost of each unit of public good provided.

Consumera's utility function is given by:

Ua(xa,G) =xa+alnG;a= 1,2,3.

Thus consumer 2's utility isx2+ 2 lnG, where I put the consumer's name as a subscript to avoid confusing it with an exponent (i.e.,x2may be mistaken forx-squared).

Suppose the government assesses each consumeraliable for a fractionaof the cost of each unit of the public good. Thus consumeracan "buy" the public good for a priceaper unit.

1. (i)Find each consumer's demand for the public good.
2. (ii)Note that the same amount of public good will be available to each consumer. Hence in equi- librium all consumers must demand the same amount. Use this to write down three equilibrium conditions, where each condition relates two of the three's.
3. (iii)Also in equilibrium, the prices paid by the three consumers must add up to the cost of the public good. Use this to write down a further condition.
4. (iv)Find the equilibrium fractionsa, and the equilibrium quantity of the public good supplied.
5. (v)Suppose the economy has total resources enough to produce a combined 20 units ofXandG.Recall that the PPF is linear withMRT=1. So the GDP of this economy is 20. Suppose also that this GDP is distributed such that consumers 1 and 2 each have 5 units of income, while 3 has 10 units of income. Write down the budget equations of the three agents, and find how Walras' Law applies to your calculations above.