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Demand Functions Worksheet

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Assignment 3 Econ 3030 Year 2022 1. Consider the following demand functions, which are the result of maximizing utility where the person’s preferences can be represented by a Cobb-Douglas function. 2 π‘₯π‘₯1 = 𝐡𝐡/𝑃𝑃1 3 1 π‘₯π‘₯2 = 𝐡𝐡/𝑃𝑃2 3 a) Use logarithms to show that the demand elasticity (price elasticity of demand) is equal to 1 for both π‘₯π‘₯1 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘₯π‘₯2 . b) Use logarithms to show that the income elasticity is equal to 1 for both π‘₯π‘₯1 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘₯π‘₯2 . c) What is the cross-price elasticity of good 1 with respect to the price of good 2? 2. Consider the function 𝑄𝑄 = 𝐾𝐾 0.4 𝐿𝐿0.7 , where each variable is a function of time. a) Write this in log form. b) if capital grows at 2% per year and labour grows at 1% per year, what is the growth rate of Q? 3. Exports, X, are growing at 6%. Imports, M, are growing 7%. At time 0, exports are 700 and imports are 800. a) Calculate the growth rate of net exports, NX, where NX = X – M b) Interpret your result. What does the number you found mean? 4. Challenge level: (though actually very quick) The elasticity of substitution is defined as percentage change in relative quantities resulting from a percentage change in the (inverse) relative π‘₯π‘₯ prices, or the percent change in 1 resulting from a percent change in the relative price line 𝑝𝑝1 /𝑝𝑝2 . π‘₯π‘₯2 Consider the ratio of first order conditions for a Cobb-Douglas utility optimization problem: π‘₯π‘₯2 π‘₯π‘₯1 = 1 𝑝𝑝1 4 𝑝𝑝2 Derive the elasticity of substitution for this case.
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