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A typical household aims to maximize its expected discounted lifetime utility function defined over consumption ct and labor nt given by: Vi = EXO BU ( C, nt). F1+ + The period utility is U (ct, nt ) = 1- "to -2 1+ and the discount factor = 148 0 denotes the rate of time preference and o is the coefficient of relative risk aversion discussed in class. Both labor parameters z and o are positive and constant. The household is endowed with one unit of time that can be used for leisure It and labor nt . One unit of labor time earns the average wage rate wt . In each period, the household consumes ct and saves in physical capital kt . Let capital's depreciation rate o equal 0 and rt denote the rate of return to physical capital. The household pays a proportional tax In only on earnings from work. A representative firm hires capital kt at rental rate it and labor at wage rate wt to produce output Ut = P(Kit, nt). . The production function is given by: F(kt, nt). = Akin , where parameter A denotes total factor productivity. The government finances its expenditure stream using tax revenue. 1. Formulate the intertemporal budget constraint of the typical household and set up its intertemporal maximization problem. Derive the first order conditions and the Euler equation. 2. Using the Euler equation, show the relationship between the rate of return r and the rate of time preference 0 in the long run where ct=ct+1 and rt = rt+1 = r