The household will be better off. The situation is illustrated in the graph below, which is not to scale in order highlight clearly what is happening. The original equilibrium is where indifference curve I1 is tangent to the black budget constraint. There are 500 gallons of gas consumed and G1 of other goods. The price of gas increases so the budget constraint rotates down to the blue budget constraint. In the absence of any government compensation, the consumer would reduce gas consumption to 400 gallons of gas and G2 of other goods. With the $50 in compensation, however, a new budget constraint appears, which is illustrated with the green line in the graph. It passes through the original equilibrium point but with the same slope as the blue budget constraint. The slopes are the same because the new price ratio is in effect. The new budget constraint goes through the old equilibrium point because, with the compensation, the consumer is able to buy his original bundle of gas and other commodities. Now, however, with the increase in gas prices, the consumer is able to achieve a higher indifference curve than was possible originally, by consuming at the point of tangency between I3 and the green budget constraint.

This is an example of the Slutzky effect. The point is that by compensating the consumer on the basis of his original gas consumption, the consumer is over-compensated. He is actually made better off because he can now achieve an indifference curve that was not originally attainable. If the government had wanted to leave him exactly as well off as before, it would only have provided compensation that was sufficient for the new budget constraint to be tangent to the original indifference curve. While, for clarity, this is not illustrated in the graph, it would be achieved by shifting the green budget constraint back to such a point of tangency.

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