Question

Governments can intervene through the social security system and one aim might be to ensure that pensioners have sufficient money to purchase goods. Assume that pensioners receive a social security income of I and no other income, and they consume two goods, X1 and X2, where X2 is a composite good (P2 = 1). Tastes of the average pensioner are captured by the utility function: U(X1,X2) = x1x2. (a) Determine the utility maximising bundle of goods that the average pensioner will buy (xi and x2 in terms of I and P). (4 marks) (b) Suppose that initially P1 = 1 and 1 = 2000, but then P1 rises such that P1 = 2. Find the original bundle. Assuming that the government wants to keep the pensioner's cost of living constant (ensuring the original bundle can be bought after the price change), determine how much compensation pensioners would need after the price change. How would the compensation change pensioners' behaviour in terms of the quantity they buy of each good? Is the change as you would expect? (6 marks) (c) If, instead, the government used Hicksian compensation, find the new optimal bundle of goods and calculate the amount of compensation that would be paid. How would the amount paid compare with your answer to part (b)? (8 marks) (d) What would need to be true for the agent's preferences, such that the two forms of compensation were identical to each other? (2 marks)

Answer 1

a) Let the price of good 1 be P₁. It is given that P₂ = 1 and income = I since they receive only the social security income and no other income.

The budget constraint of the consumer can be written as follows:

(P₁ X x₁) + (P₂ X x₂) = I

P₁x₁  + x₂ = I ( Since it is given that P₂ = 1) ......(1)

Utility of a consumer is maximised at the point where the indifference curve of the consumer is tangent to the budget line of the consumer that is the point at which the slope of indifference curve or the utility function is equal to the slope of the budget line. Slope of utility function is the MRS which is the Marginal Rate of Substitution and the slope of budget line is given by the ratio of the prices of the two goods that is P₁/P₂. So, the utility is maximised at the point where,

MRS = P₁/P₂

Marginal Utility of good 1 that is x₁ / Marginal Utility of good 2 that is x₂ = P₁/P₂

Marginal Utility of a good can be found out by differentiating the utility function with respect to that good keeping the other good constant. So, the above equation can be written as,

x₂ / x₁ =  P₁/P₂

x₂ / x₁ =  P₁   ( Since it is given that P₂ = 1)

  x₂ = x₁ P₁   .....(2)

Putting (2) in (1),

P₁x₁  + x₂ = I

P₁x₁  + P₁x₁ = I

2P₁x₁ = I

  x₁ = I / 2P₁  is the answer.

Now, putting the value of x₁ in (2),

  x₂ = x₁ P₁  

  x₂ = (I / 2P₁) x P₁

x₂= I / 2 is the answer.

So, the utility maximising bundles of an average pensioner are x₁ = I / 2P₁ and  x₂= I / 2.