4. A consumer maximises his/her intertemporal consumption, i [C₁+ i = & C²+i] - (1+ p)i subject to the following life-long budget constraint, max Ct+i i=0 (1+r)A, + T=0 ¹, a > 0, Ex[Yc+i] Et[Ct+i] (1+r)i (1+r)" Σ Σ t=0 where p is the time preference rate, the real interest rate, E[-] the expectation operator based on the information available at time=1, C, the consumption for period , Y, the labour income for period t, and A, is the wealth at the beginning of period 1. (a) Show and explain the Euler equation relating C, to expectations concerning Cr+1 (b) Show and explain the permanent income of C, if the time preference rate is equal to the real interest rate. (c) Continued with (b), show that C = rA +Y, if Y, follows a pure random walk. Explain the results

Consider the linear model Y; = B + B1X1¡ + B2X2i + u; where error term and where E (u;|X1i, X2;) = 0, observations (Yi, X1i, X2;) are independent and identically distributed, 0 < E (Y4) < ∞, 0 < E (X;) < o, 0 < E(X) < ∞ and where X2; OLS estimators ß, and B, of the coefficients B, and B, unbiased and is the X. Are the consistent? Explain.

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Suppose that Marie produces milk q using her own labor l and cattle k using the production functionq = f(k, l) = k2/3ℓ1/3Although Marie does not need to pay anyone to use either input, the opportunity costs of labor and cattle are w = 1 and v = 16, respectively, and P is the price of milk. a) Set up Marie’s short run profit maximization problem and find her short run supply curve q(P). b) Now consider Marie’s problem in the long run, where her stock of cattle may vary. Set up her long run cost minimization problem and find her labor ℓc(q) and capital kc(q) demands and cost function C(q). c) Set up Marie’s long run profit maximization problem and find her long run supply curve q(P).