EECM 3714 Lecture 10: Unit 10 Returns to scale, homogeneity and partial elasticities Renshaw, Ch. 17 21 April 2023 OUTLINE • Returns to scale • Homogeneity • Partial elasticities RETURNS TO SCALE (RTS) • Consider the production function π = π(πΎ, πΏ) • Returns to scale is the property of the production function that tells us what happens to output if both inputs are increased by the same proportion • E.g.: RTS tells us what happens to q if both K and L are doubled • Three possibilities: • Constant returns to scale: doubling L and K doubles q • Increasing returns to scale: doubling L and K more than doubles q • Decreasing returns to scale: doubling L and K less than doubles q • Page 558-571 RTS AND COBB-DOUGLAS PRODUCTION • Suppose that production is Cobb-Douglas, π = π΄πΎ πΌ πΏπ½ • Find RTS by summing πΌ, π½: • πΌ + π½ = 1 → constant returns to scale • πΌ + π½ > 1 → increasing returns to scale • πΌ + π½ < 1 → decreasing returns to scale • Don’t confuse RTS with diminishing marginal productivity • Recall that we show that a production function exhibits decreasing MP by showing that its second derivative is negative HOMOGENEOUS FUNCTIONS • What if production is not Cobb-Douglas? How does one determine RTS? • Determine if function is homogeneous; then determine the degree of homogeneity • A function π π₯, π¦ is said to be homogeneous if: • π ππ₯, ππ¦ = ππ π(π₯, π¦), where π is just a (any) constant • π is then said to be the degree of homogeneity • A production function π = π(πΎ, πΏ) is homogeneous if: • π ππΎ, ππΏ = ππ π πΎ, πΏ = ππ π • If production is homogeneous, then value of r is used to infer RTS • π = 1 → constant RTS • π > 1 → increasing RTS • π < 1 → decreasing RTS . Page 561-570 TESTING FOR HOMOGENEITY • Cobb-Douglas production: π = πΎ πΌ πΏπ½ • Increase K and L by π: ππΎ πΌ ππΏ π½ = ππΌ πΎ πΌ ππ½ πΏπ½ = ππΌ+π½ πΎ πΌ πΏπ½ = ππΌ+π½ π • This function is homogeneous; degree of homogeneity is πΌ + π½ • Linear production: π = ππΎ + ππΏ • Increase by π: π π πΎ + π π πΏ = π ππΎ + ππΏ • This function is homogeneous; degree of homogeneity is 1 = constant RTS • Log production: π = π ln πΎ + π ln πΏ • Increase by π: π = π ln ππΎ + π ln ππΏ • = π ln π + ln πΎ + π ln π + ln πΏ = π + π ln π + π ln πΎ + π ln πΏ • This function is not homogeneous • Test for homogeneity: multiply K and L by π; try to factor the π out. • Self-check: is π = πΌπΎπ½ + 1−πΌ πΏπ½ 1 ΰ΅π½ homogeneous? EXAMPLE Suppose that a firm’s production function is π = 10πΎ 0.3 πΏ0.8 . a. Is this production function homogeneous? b. Does this function exhibit increasing, decreasing or constant returns to scale? c. Does this production function exhibit decreasing returns to capital and labour (i.e. diminishing marginal productivity)? SOLUTION a. Homogeneity: • 10 ππΎ 0.3 ππΏ 0.8 = 10π0.3 πΎ 0.3 π0.8 πΏ0.8 = 10π1.1 πΎ 0.3 πΏ0.8 = 10π1.1 π • This function is homogeneous: π = 1.1 b. Returns to scale: • π = 1.1 βΉ 1.1 > 1 → increasing returns to scale • OR: since the function is Cobb-Douglas: πΌ + π½ = 0.3 + 0.8 = 1.1 > 1 → increasing RTS c. Decreasing returns to labour: • π2 π ππΏ2 = −1.6πΎ 0.3 πΏ−1.2 . This is negative for all values of K and L πΎ ≥ 0, πΏ ≥ 0 d. Decreasing returns to capital: • π2 π ππΎ2 = −2.1πΎ −1.7 πΏ0.8 . This is negative for all values of K and L πΎ ≥ 0, πΏ ≥ 0 EXAMPLE 2 Suppose π = 10[0.5πΎ 0.5 + 0.5πΏ0.5 ]0.5 . Does this function exhibit increasing, decreasing or constant returns to scale? Solution: • First, check if function is homogenous ... • 10[0.5 ππΎ 0.5 + 0.5 ππΏ 0.5 ]0.5 = • βΉ 10 π0.5 × 0.5πΎ 0.5 + 0.5πΏ0.5 10[0.5 π0.5 × πΎ 0.5 + 0.5 π0.5 × πΏ0.5 ]0.5 0.5 = 10π0.25 × 0.5πΎ 0.5 + 0.5πΏ0.5 • Function is homogenous and degree of homogeneity is π = 0.25 • Since 0.25 < 1, function exhibits decreasing returns to scale 0.5 = π0.25 π PARTIAL ELASTICITIES • Recall that if π¦ = π(π₯), then π π¦ = ππ¦ ππ₯ × π₯ π¦ • Now suppose that π§ = π(π₯, π¦). Now the partial elasticities of z w.r.t. x and y are: ππ§ π₯ ππ§ π¦ • ππ₯π§ = ππ₯ × π§ ; and ππ¦π§ = ππ¦ × π§ • Note that these two elasticities can also be written as: • • ππ§ ππ₯ π§ ÷ π₯, i.e. marginal function (w.r.t. x) divided by average (w.r.t. x) ππ§ ππ¦ π§ ÷ π¦, i.e. marginal function (w.r.t. y) divided by average (w.r.t. y) PARTIAL DEMANDELASTICITIES • Suppose that demand is π π = π(π, ππ§ , π¦), where ππ§ , π¦ are the price of another product and the income of consumers • The partial demand elasticities are: • price elasticity of demand, π π • cross-price elasticity of demand, π π§ • income elasticity of demand, π π¦ • Do examples 17.5 and 17.6 PRICE, CROSS PRICE AND INCOME ELASTICITY OF DEMAND Price elasticity of demand is: • ππ = ππ ππ × π π Income elasticity of demand is: • ππ¦ = ππ ππ¦ × π¦ π Interpretation: Interpretation: • If π π > 1 → price elastic demand • If π π¦ > 0, the good is a normal good • If π π < 1 → price inelastic demand • If π π = 1 → unit elastic demand Cross-price elasticity of demand is: • ππ§ = ππ πππ§ × ππ§ π Interpretation: • If π π§ < 0, the two products are complements • If π π§ > 0, the two products are substitutes • If 0 < π π¦ < 1, the good is a necessity • If π π¦ > 1, the good is a luxury • If π π¦ < 0, the good is an inferior good EXAMPLE 1 Suppose that the demand function is ππ = 1000 − 5π − ππ§2 + 0.005π¦ 3 and π = 15; ππ§ = 20; π¦ = 100. Find and interpret the • Price elasticity of demand • Cross-price elasticity of demand • Income elasticity of demand SOLUTION, 1 π π = 1000 − 5π − ππ§2 + 0.005π¦ 3 . π = 15; ππ§ = 20; π¦ = 100. • So, ππ = 1000 − 5 15 − 202 + 0.005 100 • ππ = ππ ππ π π × = −5 × 15 5525 3 = 5525 = −0.014 • π π = −0.014 = 0.014 < 1 → demand is price inelastic • ππ§ = ππ πππ§ × ππ§ π = −2ππ§ × 20 5525 = −2 20 × 20 5525 = −0.145 • −0.145 < 0 → the two products are complements • ππ¦ = ππ ππ¦ π¦ π × = 0.015π¦ 2 × 100 5525 = 0.015 1002 × • 2.715 > 0 → the product is a normal good; • 2.715 > 1 → the product is a luxury 100 5525 = 2.715 EXAMPLE 2 Suppose ππ = 10 + 5Τπ + 2 ln π¦ − 2π 0.1ππ§ . Also suppose that π = 2, π¦ = 100 and ππ§ = 3. a. Find and interpret: • The price elasticity of demand • The income elasticity of demand • The cross-price elasticity of demand SOLUTION • = 10 + 5Τπ + 2 ln π¦ − 2π 0.1ππ§ • So, ππ = 10 + 5Τ2 + 2 ln 100 − 2π 0.1×3 = 11.9394 wrong correct • ππ = ππ π ππ × π ππ = −5π−2 × 2 11.9394 = −5 23.8788 = −0.2094 • π π = −0.2094 = 0.2094 < 1 →demand is price inelastic • ππ¦ = ππ π ππ¦ × π¦ ππ 2 π¦ = × 100 11.9394 = 2 11.9394 = 0.1675 • π π¦ = 0.1675 > 0 → normal good; • π π¦ = 0.1675 < 1 → necessity • π ππ§ = ππ π πππ§ × ππ§ ππ = −0.2π 0.1ππ§ × ππ§ 11.9394 = −0.6π 0.3 11.9394 = −0.0678 • π ππ§ = −0.0678 < 0 → the two goods are complements OTHER PARTIAL ELASTICITIES Production elasticities: • Labour elasticity of production (elasticity of production w.r.t. labour): • ππΏ = ππ ππΏ πΏ π × = πππΏ . π΄ππΏ • This shows the (percentage) change in production due to a 1% change in labour input. • Capital elasticity of production (elasticity of production w.r.t. capital): • ππΎ = ππ ππΎ × πΎ π = πππΎ . π΄ππΎ • This shows the (percentage) change in production due to a 1% change in capital input. EXAMPLE Suppose production is π = 10πΎ 0.3 πΏ0.8 . a. Find and interpret the labour and capital elasticity of production. Solution • ππΏ = • πππΏ πΏ • π = πππΏ π΄ππΏ ππ = ππΏ πππΏ π΄ππΏ π = 8πΎ 0.3 πΏ−0.2 and π΄ππΏ = πΏ = 10πΎ 0.3 πΏ−0.2 = 8πΎ0.3 πΏ−0.2 10πΎ0.3 πΏ−0.2 = 0.8 • This means that if labour input increases by 1%, production will increase by 0.8%. (note that for CobbDouglas, this is equal to π½) • ππΎ = • πππΎ • ππΎ = πππΎ π΄ππΎ ππ = ππΎ πππΎ π΄ππΎ π = 3πΎ −0.7 πΏ0.8 and π΄ππΎ = πΎ = 10πΎ −0.7 πΏ0.8 3πΎ−0.7 πΏ0.8 = 10πΎ−0.7πΏ0.8 = 0.3 • This means that if capital input increases by 1%, production will increase by 0.3%. (note that for CobbDouglas, this is equal to πΌ) PARTIAL ELASTICITIES AND LOGS • Recall from ch. 9 that if π¦ = π(π₯), then π ln π¦ π ln π₯ = ππ₯ • Extending this to a multivariate function is simple: • If π§ = π(π₯, π¦), then • Page 580-582 π ln π§ π ln π₯ = π π₯, π ln π§ π ln π¦ = ππ¦ Homework 1. Consider the utility function π = πππ πππ.ππ + πππ.ππ π.π . a. Write down the degree of homogeneity (π) of this utility function. [2] b. Write down the elasticity of utility with respect to π (ππ ) for this utility function. [2] π 2. Consider the following production function: π = π π. ππ²π.π + π. ππ³π.π . 2a. Does this production function exhibit increasing, decreasing, or constant returns to scale? Briefly explain. [3] 2b. Use implicit differentiation to find the marginal rate of technical substitution, π΄πΉπ»πΊ. [2] 2c. Write down an expression for the isoquant if π = πππ. [2] 2d. Write down an expression for the capital elasticity of production, ππ² . [2] 3. The demand for fireworks is ππ = π + ππ + π π π₯π§ ππ − ππ−π.ππππ. Find and interpret the following partial demand elasticities (at π = πππ, ππ = ππ and π = πππ): 3a. Price elasticity of demand (ππ ). [2] 3b. Income elasticity of demand (ππ ). [2] 3c. Cross-price elasticity of demand (πππ ). [2] FINALLY… • Work through the examples and progress exercises in Ch. 17 • Next: Unit 11 - Integration
