Economic Optimization Chapter 2

Chapter 2 OVERVIEW   Economic Optimization Process   Revenue Relations   Cost Relations   Profit Relations   Incremental Concept in Economic Analysis

Chapter 2 KEY CONCEPTS   optimal decision   spreadsheet   Equation   dependent variable   independent variable   marginal revenue   revenue maximization   cost functions   short-run cost functions   long-run cost functions   short run   long run   total costs   fixed costs   variable costs   marginal cost   average cost   average cost minimization   total profit   marginal profit   profit maximization rule   breakeven points   incremental change   incremental profit   breakeven point   average cost minimization   multivariate optimization   constrained optimization   Lagrangian technique   Lagrangian multiplier, λ

© 2009, 2006 South-Western, a part of Cengage Learning Economic Optimization Process   Optimal Decisions Best decision produces the result most consistent with managerial objectives.   Maximizing the Value of the Firm Produce what customers want. Meet customer needs efficiently.   Greed vs. Self-interest Self-indulgence leads to failure. Customer focus leads to mutual benefit.

Revenue Relations   Price and Total Revenue Total Revenue = Price  Quantity.   Marginal Revenue Change in total revenue associated with a one ‑ unit change in output.   Revenue Maximization Quantity with highest revenue, MR = 0.   Do Firms Really Optimize? Inefficiency and waste lead to failure. Optimization techniques are widely employed by successful firms.

Where do demand equations come from?   Companies compile data over time on specific variables. (Time series analysis)   Companies compile data across different areas at a point in time. (Cross sectional analysis)

Demand Equation A firm estimates the demand for its product to be related to the price it charges (P), the amount of advertising it does (A), and the temperature (T). Gathering data over quarters from the past ten years, it runs a regression over the 40 observations and the computer spits out the following regression equation: D = P A T (12.1) (105.0) (9.9) R 2 = 0.74 Standard error of the estimate = 16 Standard error of the coefficients are in the ( ).

Interpreting the equation What percentage of variation of demand did the model explain? - 74% What other independent variables could have been included that probably would have increased the explanatory power of the model? - the prices of related goods, per capita income Based on the regression results, if the price of the good is increased by one unit, what will be the impact on demand? - demand would be expected to fall by 45.9 unites What if advertising is increased by one unit? - demand would be expected to increase by units

Demand Equation Observations Based on the sign of the temperature coefficient, would this product more likely be hot chocolate or Pepsi? - the positive sign indicates that as the temperature increases so will the demand for this good. Hence, it’s more likely to be a product such as Pepsi. In which coefficients would you have at least 95% confidence that you have a good estimate of the marginal relationship between the independent variable and the dependent variable? - each one for which the estimated coefficient is at least twice as great as the standard error of the coefficient (P and A; not T)

Predicting Demand given some economic starting points What is the point estimate of demand if P= 75, A=10, and T=70? - plug in these values of the independent variables in our regression equation and out falls D = One can predict with a 95% confidence of being correct that the demand of this good next period will lie in the range of units to units. The 95% confidence interval would be (16). This range is = on the upper end and – 32 = on the lower end.

What does the coefficient of P really mean? - that if the price of the good is increased by one unit while every other independent variable is held constant, the demand for this good will fall by 45.9 units. - you should recognize that the coefficient in a regression equation is nothing more than the partial derivative of the dependent variable with respect to a particular independent variable. For example:

The Supply side of the equation   Cost Marginal Cost Average Cost Fixed Cost Variable Cost

Cost Relations   Total Cost Total Cost = Fixed Cost + Variable Cost.   Marginal and Average Cost Marginal cost is the change in total cost associated with a one ‑ unit change in output. Average Cost = Total Cost/Quantity   Average Cost Minimization Average cost is minimized when MC = AC. Reflects efficient production of a given output level.

Goal is to minimize costs A business can find the quantity where costs are minimized by going to the point where MC = AC The typical cost curve.

© 2009, 2006 South-Western, a part of Cengage Learning

Maximizing Profit   Total Revenue = P * Q   TR= $24Q – $1.5Q 2   TC = $8 + 4Q +.5Q 2

© 2009, 2006 South-Western, a part of Cengage Learning

L. Pantuosco Course Notes © 2009, 2006 South-Western, a part of Cengage Learning (бD)/(б/P) = -45.9(бD)/(б/A) = 292.5(бD)/(б/T) = 17.8 Suppose the independent variables have the values given in part (g). What impact would there by on demand if the price of the goods were increased by 1%? - to answer this we need to use the concept of point price-elasticity of demand: e P = (бD/бP)/(P/D) = (-45.9)  (75/921.50) = this means that a 1% increase in price will \result in a 3.736% decrease in demand - compare this to part (i) where we saw that a one unit (not 1%) increase in price would lead to a decrease in demand of 45.9 units (not 45.9%).

© 2009, 2006 South-Western, a part of Cengage Learning Profit Relations   Total and Marginal Profit Total Profit (π ) = Total Revenue - Total Cost. Marginal profit is the change in total profit due to a one-unit change in output, Mπ = MR - MC.   Profit Maximization Profit is maximized when Mπ = MR – MC = 0 or MR = MC, assuming profit declines as Q rises.   Marginal v. Incremental Profits Marginal profit is the gain from producing one more unit of output (Q). Incremental profit is gain tied to a managerial decision, possibly involving multiple units of Q.

L. Pantuosco Course Notes © 2009, 2006 South-Western, a part of Cengage Learning What is the difference between a “point estimate of demand” and the “point elasticity of demand”? - compare the parts (g) and (j) - the “point estimate of demand” shows what demand would be at a particular point along the demand function (when the independent variables are all given particular values). - the “point elasticity of demand” shows the percentage change in demand from the point estimate of demand when there is one percent change in price.

L. Pantuosco Course Notes © 2009, 2006 South-Western, a part of Cengage Learning LEAST-COST COMBINATION OF INPUTS – AN EXAMPLE There are two approaches to this problem: (a) the optimal input ratio approach and (b) the Lagrangian Multiplier Approach. A firm has the production function: Q =40L.80 K.20. The prices of labor and capital are $1,000 and $2,000 respectively. What combination of L and K would produce an output level of 5800 units at the lowest total cost?