Appunti Microeconomics

Objective of profit maximization

The objective is maximizing π = p (L, L, K) - wlL - rkF => (L, K, t).

Example

Let's take a simple case. We cannot change the capital because in economic theory, there is usually this distinction between:

• Short Run: Time horizon which is short, situation in which some inputs are variable and some are fixed.
• Long Run: Time horizon which is long, situation in which all inputs are variable. So in the long run, all inputs can be changed (no constraint in the choice).

In the short run, we have π = p (L, L, k) - wl - rk. As we said, our objective is to maximize profit by choosing the optimal quantity of labor and capital.

Short run perspective

Let's take a short run perspective. We are in the situation in which we cannot change capital, with one factor (L) being fixed. Labor is our choice.

The equation for maximizing profit is:

max π y = f(L, k)

Suppose that we have an increase in k (but same L). The short run production function becomes y = f(L, k). Suppose that we have more capital (more than kf); what is the effect on y? The short run production function will depend on the quantity of capital that we are using.

Isoprofit lines

Suppose that we have our profit π = p y - wl - rk, which will depend on revenues minus variable costs (fixed costs). Isoprofit means same price! So an isoprofit is a line on which the profit is constant. We fix a level of profit π = p y - wl - rk. We are considering the possibility of getting some profit and relating labor and output in such a way that we achieve that level of profit.

p y = π + rk + wl

Objective in class 15

The objective is maximizing π = p (L, L_k) - W_LL + rK => (L^k + k^*).

Example in economics

In economics, there is usually this distinction between:

• Short Run: Time horizon which is short, situation in which some inputs are variable and some are fixed (technology).
• Long Run: Time horizon which is long, situation in which all inputs are variable. So in the long run, all inputs can be changed (no constraint in choice).

π = p (L, L_K) - W_L-rK. As we said, our objective is to maximize profits, choosing the optimal quantity of labor and capital.

Short run perspective in class 15

Let's take a short run perspective. In the short run, we can think of one factor (i.e., L) as fixed. We cannot change capital.

Max π short run production function X q = f (L, K). Suppose that we have more capital. What is the effect on q? The short run production function will depend on the quantity of capital that we are using, since production depends also on capital available.

Isoprofit lines in class 15

Suppose that we have our profit π = p q - W_LL - rK, which will depend on revenues minus variable costs plus fixed costs. Iso-profit means same price π, so an Isoprofit is a line on which the profit is constant. We give a level of profit π = p Y - W_LL - rK = 0. We are considering the possibility of getting some profit, and we try to obtain labor and output in such a way that we achieve that level of profit.

p Y = π + rK + W_LL

Variable | q = π/rK | W_L (variable). By rearranging the profit expression, we find the equation for a line, where there is an intercept and a slope:

_yL = (_π + _k) (_w) _p (intercept slope _{p p})

The meaning of this line is that, by using the quantity of labor L_k and by considering the output level y_k, if we compute the profit π = py_k - wL_L - k, we get exactly that level of profit (π_π'). So, all the points on the isoprofit line are production plans (labor, input, and output) that give us the same profit.

Suppose that we’re taking a new production level P', π' > π. When we draw the isoprofit line for a higher level of profit, this new isoprofit line will be shifted upward, affecting the intercept. The producer will prefer to stay on the highest possible isoprofit line. However, the isoprofit line is just a combination of production plans, so we have to check which production plans are feasible and which are not.

E is not feasible because we are in the technological area below the production function. C is not feasible. F is feasible because there are production plans giving us that level of profit that are also feasible. The production plan F, and also B, is feasible because we are on the production function. For the producer, it will be the same to be in F or B because they get the same profit. Of course, in B, they get the profit by using less labor and hence by obtaining less output. So, F and B are feasible decisions.

The producer has a better choice when considering a higher profit line. If able to reach a higher profit line, it means more profits. For the producer, it is the same to be in E or in D.