Question

1. Suppose that Alex must produce his own cereal, using inputs of raw materials obtained from his parent's food storage (S) and labor provided by Alex's cook- ing skills (L). Production of cereal is given by Fc(LS)-L0.5S0.5. Further. suppose Alex has a utility function Ua(ca,Ca) c505, where ca is his consump- tion of cereal and la is the amount of leisure he enjoys. He has no endowment of cereal, but is endowed with 120 hours of time (which can be sold as labor or kept as leisure) and 40 food storage cans (s) (a) (2 pts) Set up Alex's household problem (b) (2 pts) Alex creates a firm to produce cereal. Set up the firm's problem (c) (4 pts) Define a competitive equilibrium for this economy. (You may refer to the previous two parts as appropriate, rather than repeating them, but be precise.) Be sure to specify the market clearing condition for each market (d) (4 pts) Solve for the aggregate (i.e. Alex's) demand for leisure and for there should be no other cereal (in terms of input and output prices variables in the resulting equations) You should get c120p40p 2pe and 120pe+40pII 2pe (e) (4 pts) Solve the firm's problem Remember that with CRS production, this will just pin down rel ative prices, not give you solutions for C and L. You should get 4pe (f) (4 pts) The previous question gives a profit-maximizing relationship be- tween S and L (in macro, this is called the capital-to-labor ratio). Use this to substitute for L in the production function, and simplify to get the supply of C as a function of input S. Also, compute the profit earned given S 2 GE Production 0.5 You should get C- ) S and 11-0. (Others can be found. but are equivalent if you substitute using the price relationship from the previous problem.) (g) (3 pts) Solve for the equilibrium prices Normalizing the price of cereal to 1 mαy be easiest. This looks messy at first, but you should get p- and p- (h) (2 pts) Solve for the equilibrium allocation You should get c*ー40,(*ー80. L* = 40. 'S* = 40, C* 40

Answer 1

a)His household problem is :

1)Max U(ca,la ) s.t.

(2) p(c ) + wL h + rA h<=F(L, A) + w(L m 1 + L m 2 ) + rA m

(3) L = L f 1 +L f 2 + L h

(4) A = A f + A h

(5) E A ' A f + A m

(6)E L i = L f i + L m i + l i , i0{1,2}

(7) ci , l i , L f i , L m i , A f , A m $ 0, i0{1,2}

(B)The firm's problem is

Equation (1) is a household utility function

Equation (2) is a conventional budget constraint: cash expenditures on consumption, hired labor and rented land cannot exceed cash revenues from farming, market labor, and land rented out.

Equations (3) through (5) define resource constraints: labor use on the farm is household labor used on the farm plus hired labor; land use on the farm is owned land used on the farm plus hired land; the household's land endowment is used on its own farm or rented out, and each individual's time endowment equals their labor use on the farm, plus market labor time, plus leisure time. Substituting equations (3)-(5) into (2) we find:

(7) p(c1 + c2 ) + w(l1 +l 2 ) <= A % w(EL1 + EL2 ) + rE A

(8) A ' F(L,A) & wL & rA

(9) ci , l i , L, A >= 0, i{1,2}

Equation (7) is called the “full income” constraint - the value of consumption cannot exceed the value of the household's endowment plus farm profits. The household's problem is now to maximize (1) (now with respect to L, A, c and l) subject to (7) - (9)