Question

In: Economics

1. Consider a farmer that grows hazelnuts using the production q = AL2/3 , where q...

1. Consider a farmer that grows hazelnuts using the production q = AL2/3 , where q is the amount of

hazelnuts produced in a year (in tonnes), L represents the number of labor hours employed on the farm during the year, and A is the size of orchard which is fixed. The annual winter pruning of the orchard cost $1000, and is already paid by the farmer. The hourly wage rate paid for labor is $12.

  1. Derive the marginal cost function.
  2. Write the farmer’s profit maximization problem.
  3. Derive the farmer’s short run supply equation using your answer for f.
  4. Suppose the rainfall in the region where the farmer operates in was far lower than what was expected at harvest. This increased the productivity of labor by 50%. Derive the new marginal cost function assuming the wage rate remains unchanged.

Solutions

Expert Solution

Solution:

We have production function as: q = A*L2/3

So, we can write as follows: L = (q/A)3/2

The annual cost of orchard pruning is the fixed cost for given situation (FC = $1000) as it is independent of the output level; and as total wage cost changes with output (by way of labor hours), it indicates the variable cost. So VC = 12*L. VC = 12*(q/A)3/2

As we know, total cost, TC = Fixed cost, FC + Variable cost , VC

TC = 1000 + 12*(q/A)3/2 , where A is fixed

a) Marginal cost, MC = = (3/2)*(12/A3/2)*q3/2-1

So, MC = (18/A3/2)*q1/2

b) Farmer's profit, W = total revenue - total cost

Total revenue = price*quantity = P*q (P is the price)

So, W = P*q - (1000 + 12*(q/A)3/2)

Farmer's maximization problem is simply to maximize his/her profits, W here.

c) Now, profit is maximized where the first order condition (FOC): = 0 is satisfied

= P - (18/A3/2)*q1/2

So, with solving the FOC as mentioned above, we get: P = (18/A3/2)*q1/2

This derived equation is the short run supply equation (found using the above part).

Rahul Sunny answered 1 year ago
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