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The consumer’s problem

The consumer’s problem is 

max {c,l,ns} u(c, l) 

subject to c = wns + d − T l + n s = 1. 

Suppose that the utility function is u(c, l) = c α l 1−α , where 0 < α < 1 is the parameter of the utility function. 

Please find the goods demand function c(w, d, T), leisure demand function l(w, d, T), and the labor supply function n s (w, d, T). Note that these three functions contain the parameter α. Are both c and l normal goods for this consumer? Explain. Hint: uc = αcα−1 l 1−α , and ul = (1 − α)c α l −α . (20 points)

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