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Matematika ekonomi


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Matematika ekonomi

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Application of differentiation l.jpg

Application of Differentiation


Derivatives of logarithmic functions l.jpg

Derivatives of Logarithmic Functions

  • Find dy/dx

    1.

    2.

    3.

    4.


Derivatives of exponential functions l.jpg

Derivatives of Exponential Functions

Find dy/dx

1.

2.

3.


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Applied Maxima and Minima

  • Maximizing Revenue

  • Minimizing Average Cost

  • Profit Maximization

  • Elasticity of Demand


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Maximizing Revenue

1. The demand equation for a manufacturer’s product is p = 20 - 0.25q, 0  q  80

where q is the number of units and p is the price per unit. At what value of q will there be maximum revenue? What is the maximum revenue?

2.The demand equation for a monopolist’s product is p = -5q + 30. At what price will revenue be maximized?


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Minimizing Average Cost

1.A manufacturer’s total cost function is given by c = q2/4 + 3q + 400

where c is the total cost of producing q units. At what level of output will average cost per unit be a minimum?

2.A manufacturer finds that the total cost c of producing a product is given by the cost function c = 0.05q2 + 5q + 500

At what level of output will average cost per unit be a minimum?


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TUGAS

1. For Monopolist’s product, the demand function is p = 72 - 0.04q and the cost function is c = 500 + 30q. At what level of output will profit be maximized? At what price does this occur, and what is the profit?

2. For a monopolist’s product, the demand equation is p = 42 - 4q, and the average cost function is = 2 + 80/q. Find the profit maximizing price


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Profit Maximization

3. Suppose that the demand equation for a monopolist’s product is p = 400 - 2q and the average cost function is = 0.2q+4+400/q, where q is number of units and both p and

are expressed in dollars per unit

a.Determine the level of output at which profit is maximized

b.Determine the price at which maximum profit occurs

c.Determine the maximum profit


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Elasticity of Demand

  • Economists measure how a change in the price of a product will affect the quantity demanded

  • The ratio of the resulting percentage change in quantity demanded to a given percentage change in price:

    % change in quantity

    % change in price


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Continue …...

  • If p = f(q) is a differentiable demand function, the point elasticity of demand denoted by:


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Continue …...

1. When || > 1, demand is elastic

2. When || = 1, demand has unit elasticity

3. When || < 1, demand is inelastic


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Continue …...

  • Example 1

    a. Find the point elasticity of demand for the demand function p = 1200 - q2 at q = 10.

    b. Determine the point elasticity of demand equation q = p2 - 40p + 400 (where q > 0) at p = 15.


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Elasticity and Revenue


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Continue …...

  • If demand is elastic, then  < -1,

    so 1 + 1/  > 0

     total revenue increases if demand elastic

  • If demand is inelastic, then  > -1,

    so 1 + 1/  < 0

     total revenue decreases if demand

    inelastic