Suppose there is a family farm with l acres of land The outp

Suppose there is a family farm with l acres of land. The output from the farm is y = zf(l) = zl^a for a element of (0,1). The family can choose to have a certain number of children k. Children are not productive, they simply consume resources. Each child requires times units of consumption to raise. The family gets utility from consumption left over after raising kids (c), as well as some intrinsic utility from having children. Suppose that utility is given by u(c, k) = log(c) + beta log(k) Pretend that you can have fractional children, so that the choice of k is continuous (any k > = 0) rather than discrete (k element of {0,1,2,...}). Write down the budget constraint for the family. It should relate the choices of c and k and will also include l and x. Write down the maximization problem of the family. Find the optimal choices of c and k. What is the relationship between the optimal c and k. What is the relationship between y and the optimal k Now imagine a society of N such families and and a total amount L of land. The children raised today become the families of the next generation so that N\' = (k/2)N. How does this map into the Malthusian framework from lecture Find the equilibrium values for c, y, l, and N. How do these vary with the preference for children, beta and overall productivity z

Solution

(a) Budget constraint shows all possible combination of goods and services which consumer can consume with his or her given income during a particular period of time. A consumer cannot consume those combination of bundles which are above budget line. Budget constraint is given by:

P_X.X + P_Y.Y = M

where, P_X is the price of commodity X

X is the quantity of commodity X

P_Y is the price of commodity Y

Y is the quantity of commodity Y

According to the question, budget constraint will be:

x.k + c.l = y

where, x = Unit of consumption by each child

k = No. of childrens

c = Utility family get after consumption of their childrens

l = acres of land

y = total output of farm

This budget constraint shows that total output of farm remains fixed under which all families have to attain their combination of goods which provide them maximum benefit. Consumption of goods by childrens is denoted by x.k and remaining amount of output is used by family members. Total consumption of goods by childrens and other family members must equal to the total output of the farm.

Yes, the above situation includes c, k, l and x because all these factors affect the consumption of farm product.