Consider a representative consumer that has preferences over bundles of consumption goods C, and leisure L. The consumer's preferences satisfy the properties: more is preferred to less; preference for diversity; normality. The consumer is endowed with total hours h. Let the consumer's real wage per hour worked in terms of the consumption good be w. Real firm profits, which are distributed to consumers as dividends, are denoted . The real lump-sum taxes paid by consumers to the government are T < .

(a) Suppose now that in addition the government provides a proportional subsidy s on real wage income to consumers. What is the effect of the proportional subsidy on the consumer's choices of consumption and leisure, as well as on their welfare? Explain the results in terms of substitution and income effects.

(b) Suppose now that, instead of (a), the consumer faces a progressive tax system on their labour income earnings (wN^s ): for the first y units of their labour income the consumer pays a proportional tax t₁ on each unit of real wage income, and for real wage income above y the consumer pays a proportional tax t₂ > t₁ on each unit of real wage income. Write down the consumer budget constraint (make sure to specify what happens below and above y). Draw the consumer budget constraint on a labeled graph. Show the optimal consumption choice for a consumer with labour income above y.