Pocket money, £15, may be spent on either pool (60p per hour) or skating (£1.20 per hour).

(a) Write down the equation of the budget constraint. Graph the constraint.

(b) Show by calculation and graphically how the budget constraint changes when:

(i) The price of pool increases to 75p, the other variables do not change.

(ii) The price of skating drops to 90p, the other variables do not change.

(iii) Pocket money decreases to £10, the other variables do not change.

Step-by-Step

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(a) Let x = number of hours of pool, P_X = 60; y = number of hours skating, P_Y = 120; M = 1500, the income limit. The budget constraint is given by the equation

x P_X + yP_Y = M → 60x + 120y = 1500

Write the budget constraint in the form y = c + mx→y = 12.5 − 0.5x.

To plot this line in Excel, determine the range of values of x for which x and y are both positive. What we actually require is the x-intercept to plot the graph for values of x between x = 0 and x = x-intercept.

The budget constraint or budget line cuts the x-axis at y = 0, hence

0 = 12.5 − 0.5x → x = 25

Therefore, set up a table in Excel to calculate y (use the formula command) for values of x (at equal intervals), over the range x = 0 to 25. See Figure 2.45.

(b) (i) Pool increases to 75p per hour, so fewer hours of pool are affordable. This will cause the x-intercept to decrease towards the origin or, in other words, the budget line pivots downwards from the unchanged vertical intercept. The new budget line is given by the equation

x P_X + yP_Y = M → 75x + 120y_{bi} = 1500

or y_{bi} = 12.5 − 0.625x

Again, use Excel to calculate a table of points and plot this budget line along with the original budget line. See Figure 2.46.

**Note:** The subscript ‘bi’ (part (b), section (i) of question) attached to the variable y is used only for referencing the new budget line in the figure. The same applies to ‘bii’ and ‘biii’ below.

(ii) Skating decreases from 120p to 90p per hour, so more hours of skating are affordable. This will cause the y-intercept to increase away from the origin, that is, the budget line pivots upwards from the unchanged horizontal intercept. The new budget line is given by the equation

x P_X + yP_Y = M → 60x + 90y_{bi i} = 1500

or y_{bi i i} = 16.67 − 0.67x

Again, use Excel to calculate a table of points and plot this budget line along with the original budget line. See Figure 2.47.

(iii) This time income decreases from £15 to £10, so less of everything is affordable.

The new budget constraint drops, but runs parallel to the original. The new budget constraint is given by the equation

x P_X + yP_Y = M → 60x + 120y_{bi i} = 1000

or y_{bi i i} = 8.33 − 0.5x

Again, use Excel to calculate a table of points and plot this budget line along with the original budget line. See Figure 2.48.

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