( 2 0 Marks ) Answer the following:

i . Present graphically production plan, feasible production plan, efficient production plan, and optimal production plan.

ii . Given the production function f ( x 1 , x 2 ) = x 1 0 . 5 x 2 0 . 5 , calculate the profit maximizing demand and supply functions, and the profit function.

i. Graphically representing production plans involves plotting the different types of plans on a graph.

• A production plan is a representation of the quantities of all inputs used to produce a given output. This can be represented on a graph with the quantity of one input on the x-axis and the quantity of another input on the y-axis. The production plan is then a point on this graph.
• A feasible production plan is a production plan that is possible given the constraints of the production process. This can be represented on the same graph as the production plan, but it is a region rather than a point. The feasible production region is the set of all points that represent feasible production plans.
• An efficient production plan is a production plan that maximizes output given the inputs. This can be represented on the same graph as the feasible production region, but it is a point on the boundary of the feasible production region.
• An optimal production plan is the most efficient production plan given the constraints of the production process. This can be represented on the same graph as the efficient production plan, but it is a point on the boundary of the feasible production region that also satisfies the conditions of the production function.

ii. Given the production function f(x1,x2)=x10.5×20.5, we can calculate the profit maximizing demand and supply functions, and the profit function.

The profit function is given by the difference between total revenue and total cost. Total revenue is the price times the quantity sold, and total cost is the cost of production.

The profit maximizing demand function is the quantity of the good that consumers are willing to buy at the profit maximizing price. This is found by setting the derivative of the profit function with respect to price equal to zero and solving for quantity.

The profit maximizing supply function is the quantity of the good that producers are willing to supply at the profit maximizing price. This is found by setting the derivative of the profit function with respect to cost equal to zero and solving for quantity.

The profit function is then the difference between total revenue and total cost at the profit maximizing quantities.

Without specific values for price, cost, and the quantities of the inputs, it’s not possible to provide specific functions or values. However, the general approach would be to take the derivatives of the profit function with respect to price and cost, set these equal to zero, and solve for the quantities of the inputs.