Maximise: U = f(x, y)
Subject to: x.Px + y. PY < M
Equation 2.16 expresses the consumer’s objective while his constraint to fulfil this is expressed by 2.17. Here, the utility maximising bundle comprises quantities x and y of goods X and Y available to the consumer at respective prices Px and Py. Required expenditure on the two goods is x.Px + y. PY which the consumer has to finance from his budget, M. Constraint 2.17 requires consumer’s expenditures to be less than or equal to M . But the utility maximising consumer has to make the best of the available budget. Hence the constraint x.Px + y. PY ? M is converted to x.Px + y.PY = M . The mathematical version of the consumer’s problem, also called the mathematical model, can now be recast as below:
Maximise: U = f(x, y)
Subject to: x.Px + y.PY = M
It is to be noted that the problem is no longer one of simple maximisation. Had it been so, use of simple calculus would have been enough. The problem here is that of constrained maximisation. We therefore need to convert it into one of simple maximisation first. The technique of Lagrange’s multiplier comes handy for the purpose. Employing the technique, we merge the constraint into the objective function and apply calculus thereafter.
The condition of consumer’s equilibrium thus works out as in Equation 2.18.
MUx / MUy = Px/Px
We explain here the Lagrange’s Method of constrained maximisation. It requires familarity with partial differentiation. Re-writing Equation (2.16) as
The right hand side of Equation 2.18 gives the price ratio of the two goods while its left hand side gives the MRSx,y.
For diagrammatic representation of the consumer’s equilibrium, we require the plot of the budget line and the indifference map. The latter consists of a number of indifference curves, one above the other, representing distinct levels of utilities available to the consumer. Figure 2.32 shows the Indifference Map.
Equation (2.23) gives the required condition of equilibrum of the concumser. Here,
The former, the budget line, is the locus of possible combinations of quantities of two goods that can be bought out of a given budget (M) at the respective prices Px and PY. The set of points on the budget line is known as the budget set.
Figure 2.33 shows the budget line of the consumer. Price of Y remaining the same, a fall in price of commodity X decreases the price ratio causing an outward pivotal shift in the budget line, and a rise in it increases it causing an inward pivotal shift in the budget line as shown in Figure 2.34.
If income increases, or if prices of both the goods fall in the same proportion, the budget line undergoes a parallel shift upwards (2.35a). The opposite is the effect of a decrease in income or a simultaneous increase in prices of the two goods by the same proportion (2.35b).
Consumer’s equilibrium can be shown diagrammatically as in Figure 2.36. Consumer’s budget line (Figure 2.33) and his indifference map (Figure 2.32) are both superimposed in Figure 2.36. The highest attainable indifference curve is U” which the budget line AB touches at point e. The equilibrium basket had by the consumer is (xe, ye).
Given the prices of the two goods as Px =100 and Py = 50 (From the equation of the budget line), we have
2 = 100/50 = 2.
(MUX/MUY = PX/PY)
That is, MRSX, Y = PX/PY for all baskets represented by the utility function or by the budget line. This also indicates that the budget line and the utility function must both coincide (note that the utility function is linear, indicating that the two goods are substitutes). There exist infinitely many solutions, that is any basket on the budget line is as good as any other on it (Fig. 2.37).
Given the prices of the two goods as Px = 100 and PY = 50 (from the equation of the budget line given), we have
PX/PY = 100/50 = 2
4 ? 2 (MRSx y < PX/Py)
(This time, IC is steeper than the budget line) Again, utility function being linear, the goods being substitutes, (30, 0) would optimise consumer’s utility (Fig. 2.39).