Question

Tomato yields

Number of plants 3 4 5 6 7 8 Pounds of tomatoes 42 54 65 75 84 92

Chris runs a CSA (“community sponsored agriculture”) farm, and they are trying to determine the optimal amount of “investment” in tomato seedlings. The table above shows the plot’s (expected) yield as a function of the number of tomato plants.

1. (a) Tabulate the MPK, in terms of pounds of tomatoes, as a function of the number of tomato plants.

2. (b) Assumetheinterestrateis7%(r=0.07),thepriceofaseedlingis$1,andtomatoessellfor$0.10/pound.

   Tomato plants are annuals (they live only one year) so d = 1. Calculate the user cost.

3. (c) Find the optimal amount of investment (the number of seedlings to plant).

4. (d) Repeat the calculation with:

   1. (i) A 5 percentage point increase in the interest rate, to 12%.

   2. (ii) A drop on the price of tomatoes, to $0.09/pound (r = 7%).

   3. (iii) A drop in the price of seedlings, to $0.90 (r = 7% and tomato price = $0.10).

   4. (iv) The introduction of a new variety of tomatoes with a 10% higher yield, regardless of the number

      planted (r = 7%, tomato price = $0.10 and seedling price = $1).

Answer 1

a).

Here we have given the “Numbers of Plants” and their corresponding “Pounds of tomatoes”. So, the marginal productivity of capital measures the additional production from additional use of capital. So, the following fig shows the MPK for each unit of plants hired.

b).

Here the interest rate is “r=0.07”, depreciation rate is “d=1” and the price of a seedling is “Pk=$1”. So, the “user cost of capital” is given below.

=> User Cost of Capital = (r+d)*Pk = 1.07*$1 = $1.07.

=> User Cost of Capital = $1.07.

c).

Here given the MPK for each unit of plant used. So, the return of each unit of plant used is “P*MPK”, where “P” is the price of tomato per pound. The above table shows the return of each unit of plant used. Here for “K=5” the return is “$1.1 > user cost = $1.07”, => additional use of plant increase the total return. On the other hand for “K=6” the return is “$1 < user cost = $1.07” , => additional use of plant decrease the total return. So, the optimum plant size is “K=5”.

d/i).

Let’s assume the interest rate increases to “r=0.12”, => the user cost also increases to “(r+d)*Pk = 1.12*$1 = $1.12. Now, the return of capital remain the same, as “P” and MPK remains the same. So, the optimum plant size is “K=4”, where “P*MPK = user cost of capital”.

ii).

Let’s assume the price of tomato deceases to “0.09 per pound”, => the return of capital also decreases. The following table shows the new return of capital.

Here for “K=4” the return is “$1.08 > user cost = $1.07”, => additional use of plant increase the total return. On the other hand for “K=5” the return is “$0.99 < user cost = $1.07” , => additional use of plant decrease the total return. So, the optimum plant size is “K=4”.