Problem set 4Buan 6340Linear algebra1. Real business cycles:Consider the following macroeconomic model for real business cycles taken from Dejong and Dave (2011)Chapter 3:Suppose that we have a representative household whose goal is to maximize their utility from consumptionand leisure over time:U = E0∞ Xt=0βtu(ct, lt)where E0 is the expected value at time 0, β is a parameter reflecting the household’s discount rate (timepreferences), ct is the household’s consumption at time t, and lt is the household’s leisure time at time t.For this problem to work, we have to introduce four constraints. The first constraint is that the householdproduces goods using a Cobb-Douglas production function. That is,yt = ztktαn1 t-αwhere yt represents the GDP at time t, zt represents and exogenous (i.e., cannot be optimized or changed)technology stock at time t, kt represents the gross capital stock at time t, and nt represents the amount oflabor supplied at time t. The second constraint is that total time spent working and at leisure should add tosome fixed constant. Without loss of generality we can define the total to be one (1):1 = nt + ltThe third constraint is that GDP can be spent in two ways: consuming and investing in future capital:yt = ct + itwhere it is the amount of investment at time t.kt = it + (1 – δ)ktThe last thing we need to define to make the problem solvable is the utility function. For this we are going touse a constant elasticity of substitution (CES) form. That is,u(c, l) =cφl1-φ1-ρ1 – ρ1where rho and phi are parameters representing the trade off between happiness from consuming versus leisurein the model.Your job is solve this consumer’s problem using calculus by maximizing the total utility subject to theconstraints defined above. In the end, you should obtain two equations. The first equation will reveal theconsumer’s inner-temporal choice between consumption and leisure. The second equation will reveal theconsumer’s intertemporal choice between consuming today and consuming tomorrow. If you are struggling,the answer is in Dejong and Dave (2011) Chapter 3.Programming2. Ridge regression:Write some code which perform L2 regularization. That is, solve OLS with a constraint that the L2 norm ofthe betas is less than some threshold value.minβnXi=1e2i subject torXi=1βi2 ≤ T2

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