- 1. Economic Models: Basic Mathematical Tools applied in economics
- 2. Simplified representations of reality play a crucial role in economics. Why models?
- 3. Models in Economics •A model is a simplified representation of a real situation that is used to better understand real-life situations. Create a real but simplified economy Simulate an economy on a computer Ex.: Tax models, money models •The “other things equal” assumption means that all other relevant factors remain unchanged.
- 4. Demand A relation showing how much of a good consumers are willing and able to buy at each possible price during a given period of time, other things held constant
- 5. Law of Demand • A decrease in the price of a good, all other things held constant, will cause an increase in the quantity demanded of the good. • An increase in the price of a good, all other things held constant, will cause a decrease in the quantity demanded of the good.
- 6. Demand schedule Demand curve Price Quantity Demanded 1.25 $ 8 1.00 $ 14 0.75 $ 20 0.50 $ 26 0.25 $ 32 $- $0.25 $0.50 $0.75 $1.00 $1.25 2 8 14 20 26 32 Quantity Price D The Demand Curve The demand curve slopes downward because of the law of demand
- 7. Functional Relationships • Relationship between two variables, for e.g. price and output sold, expressed in various ways Table or graph Use of equations – Quantity sold depends on the price, in other words quantity sold is a function of price. P is the independent value and Q is the dependent value p p f Q 5 200 ) (
- 8. Linear Demand Function QX = a0 + a1PX + a2N + a3I + a4PY + a5T PX QX Intercept: a0 + a2N + a3I + a4PY + a5T = 200 Slope: QX/PX = a1 =-5
- 9. Non-linear functions • For e.g. Total Revenue • TR=PQ • Marginal Revenue (Slope of Total revenue) • MR=Change in TR associated with change in Q
- 10. Tabular form Representation Q P=100-10Q TR=100Q-10Q2 AR MR 0 100 0 - 0 1 90 90 90 90 2 80 160 80 70 3 70 210 70 50 4 60 240 60 30 5 50 250 50 10 6 40 240 40 -10
- 11. Graphical Representation & Concept of Slope & Curvature TR TR Q O A B C
- 12. • Slope of TR Curve at a particular point represents MR at a particular output, i.e., change in TR for an infinitesimal change in output level • Implication of slope for any variable implies marginal value of the same variable • Curvature depends on changes in slope or changes in marginal value Changes in Slope
- 13. Changes in Curvature • Linear Curve – Marginal value constant, no change in curvature • Curve Convex to the origin – Marginal value (Slope) changing at an increasing rate • Curve Concave to the origin – Marginal value ( Slope) changing at a decreasing rate
- 14. Average and Marginal • Graphically Average value can be derived from the total value curve. • Average at a point on the Total value curve is equal to the slope of the ray from the origin to that particular point • To increase (decrease) the average value, Average value should be less (more) than the Marginal value • Average Value constant implies its equality with Marginal Revenue
- 15. Find out from Total Cost, Average, & Marginal Cost Q TC AC MC 0 20 - - 1 140 140 120 2 160 80 20 3 180 60 20 4 240 60 60 5 480 96 240 AC = TC/Q MC = TC/Q
- 16. Average Cost (AC) Q TC AC MC 0 20 - - 1 140 140 120 2 160 80 20 3 180 60 20 4 240 60 60 5 480 96 240 AC = TC/Q
- 17. Total, Average, and Marginal Cost Q TC AC MC 0 20 - - 1 140 140 120 2 160 80 20 3 180 60 20 4 240 60 60 5 480 96 240 AC = TC/Q MC = TC/Q
- 18. Total, Average, and Marginal Cost 0 60 120 180 240 0 1 2 3 4 Q T C ($) 0 60 120 0 1 2 3 4 Q AC , M C ($) AC M C
- 19. Optimization Techniques • In Economics different optimization techniques as a solution to decision making problems • Optimization implies either a variable is maximized or minimized whichever is required for efficiency purposes, subject to different constraints imposed on other variables • E.g. Profit Maximization, Cost Minimization, Revenue Maximization, Output Maximization • A problem of maxima & minima requires the help of differential calculus
- 20. Profit Maximization Q TR TC Profit 0 0 20 -20 1 90 140 -50 2 160 160 0 3 210 180 30 4 240 240 0 5 250 480 -230
- 21. Profit Maximization 0 60 120 180 240 300 0 1 2 3 4 5 Q ($) MC MR TC TR -60 -30 0 30 60 Profit
- 22. Marginal Analysis to profit maximization • Marginal Analysis requirement for profit Maximization, Marginal Revenue = Marginal Cost (MR) (MC) • Marginal Value represents slope of Total value curves, • Thus slopes of TR &TC should be equal
- 23. Conditions of Profit Maximization • MR=MC is a necessary condition for Maximization, not a sufficient one as this condition also hold for loss maximization • Sufficient condition requires that reaching a point of maximization, profit should start declining with any further rise in output, i.e. Slope of TC should rise & Slope of TR must fall after reaching the point of Maximization, • Change in MC>Change in MR
- 24. Concept of the Derivative The derivative of Y with respect to X is equal to the limit of the ratio Y/X as X approaches zero.
- 25. Rules of Differentiation Constant Function Rule: The derivative of a constant, Y = f(X) = a, is zero for all values of a (the constant). ( ) Y f X a 0 dY dX
- 26. Rules of Differentiation Power Function Rule: The derivative of a power function, where a and b are constants, is defined as follows. ( ) b Y f X aX 1 b dY b aX dX
- 27. Rules of Differentiation Sum-and-Differences Rule: The derivative of the sum or difference of two functions U and V, is defined as follows. ( ) U g X ( ) V h X dY dU dV dX dX dX Y U V
- 28. Rules of Differentiation Product Rule: The derivative of the product of two functions U and V, is defined as follows. ( ) U g X ( ) V h X dY dV dU U V dX dX dX Y U V
- 29. Rules of Differentiation Quotient Rule: The derivative of the ratio of two functions U and V, is defined as follows. ( ) U g X ( ) V h X U Y V 2 dU dV V U dY dX dX dX V
- 30. Rules of Differentiation Logarithmic Function: The derivative of a log function of X is defined as follows. d(log X)/dX= 1/X
- 31. Rules of Differentiation Chain Rule: The derivative of a function that is a function of X is defined as follows. ( ) U g X ( ) Y f U dY dY dU dX dU dX
- 32. Using derivatives to solve max and min problems Optimization With Calculus To optimize Y = f (X): First Order Condition: Find X such that dY/dX = 0 Second Order Condition: A. If d2Y/dX2 > 0, then Y is a minimum. OR B. If d2Y/dX2 < 0, then Y is a maximum.
- 33. CENTRAL POINT The dependent variable is maximized when its marginal value shifts from positive to negative, and vice versa
- 34. The Profit-maximizing rule Profit(p) = TR – TC At maximum profit dp/dQ = dTR/dQ - dTC/dQ = 0 So, dTR/dQ = dTC/dQ (1st.O.C.) => MR = MC d2TR/ dQ2 = d2TC/dQ2 (2nd O.C.) ==> dMR/dQ < dMC/dQ This means slope of MC is greater than slope of MR function
- 35. Constrained Optimization To optimize a function given a single constraint, imbed the constraint in the function and optimize as previously defined