Questions on Applications of Partial Derivatives [In Economics]

  1. Show that the rate of change of marginal utility of commodity with respect to y is equal to the rate of change of marginal utility of commodity with respect to x, where function is given by \(U = 3x^2y^2\ + \ y^2 \).
  2. The production function of a firm is given as \(Q = 8LK \ -\ L^2\ -\ K^2\ ,\ L>0,\ K>0\). Find the marginal productivities of labour (L) and capital (K)  and also show that \(L\frac{\partial Q}{\partial L}+K\frac{\partial Q}{\partial K}\; =\; 2Q\).
  3. Given the production function: \(Q = 4KL\ -\ 2K^2\ -\ L^2\). Find the maximum P with the constraint, \(L\ +K\ =\ 10\).
  4. The production function of a firm is given as \(Q\ = 4L^{3/4}K^{1/4}, L>0,\ K>0\). Find the marginal productivities of labour (L) and capital (K). Also show that \(L\frac{\partial Q}{\partial L}+K\frac{\partial Q}{\partial K}\; =\; Q\).
  5. Given the production function: \(Q = [\beta K^{-a}+\alpha L^{-a}]^{-1/a}\). Show that, \(L\frac{\partial Q}{\partial L}+K\frac{\partial Q}{\partial K}\; =\; Q\).
  6. For the production function \(X\; =\; \sqrt{aL^{2}\; +\; 2hLK\; +\; bK^{2}}\) , where X, L and K represent output, labour and capital respectively. Show that L time the marginal physical product of labour plus K times the marginal product of capital equals total product.
  7. A consumer’s utility function is given as : \(U\; =\; \sqrt{x_{1}x_{2}}\), where \(x_1 \ , x_2\) denote the quantities of two products consumed by the consumer and the prices per unit of the goods are Rs. 20 and Rs. 10 respectively. Determine the optimum level of commodities to maximise his utility and spend his total income of Rs. 640 on two goods.
  8. A consumer has the following utility function defined over \(x_1\) and \(x_2\) as : \(U\; =\; \left( x_{1}\; ,\; x_{2} \right)\; =\; a_{1}\; \log \; x_{1}\; +\; a_{2}\; \log \; x_{2}\) subject to \(a_1\ +\ a_2\ =\ 1\). Find the demand schedules for \(x_1\) and \(x_2\).
  9. And individual’s utility function for two goods is : \(U\ =\ x^{\alpha}y^{\beta}\).. It is given that \(p_x\) and \(p_y\) are the fixed prices of two goods X and Y (you need not use the second order condition). Deduce that the elasticity of demand for either good with respect to income or its price is equal to unity.
  10. Suppose there is a production function of the type: \(Z\ =\ e^{x^2\ +\ 2xy\ +\ 3y^2} \) where Z is the product and x and y are different factors of production, find the marginal product of x and y.
  11. The utility function is : \(U\ =\ 108\ – [(x-6)^2+2(y-6)^2]\) and the budget equation is: \(3x\ + \ 5y\ =\ 25\). For which values of x and y, U will be maximum U also.
  12. For Cobb-Douglas production function \(x\ =\ f(L,K) = AL^{\alpha}K^{\beta}\) , where x, L and K are the units of product, labour and capital respectively, show that there is increasing, decreasing or constant returns to scale according as \((a\ +\ b)\ >\ 1, <1\ or\ =1 \).
  13. Find the ratio of the marginal utilities of two goods, when the utility function is \(U\; =\; \left( ax\; +\; by\; +c\sqrt{xy} \right)\). Verify that the same result is obtained when the utility function is written as : \(U_1\; =\; log \left( ax\; +\; by\; +c\sqrt{xy} \right)\).
  14. Using calculus, using calculus, show that : \(MR\; =\; AR\; \left[ \; 1\; -\; \frac{1}{\mbox{E}_{p}} \right]\) and verify for the demand functions \(x\; =\; \frac{a\; -\; p}{b}\; ,\; b\; \neq \; 0\).
  15. The demand D of passenger automobiles is given by : \(D = 0.90\ I^{1.1} p^{-0.7}\) where I is the income and p is the price per car. Find (i) the price elasticity of demand and (ii) the income elasticity of demand.
  16. If \(x_1\) and \(p_1\) are the demand and price of tea; \(x_2\) and \(p_2\) are demand and price of coffee and the demand functions are given as ; \(x_1\ = p_1^{1.7} p_2^{0.6} \) , \(x_2\ = p_1^{0.4} p_2^{-0.8}\) . Calculate the two cross elasticities of demand and point out if the commodities are competitive or complementary.
  17. If \(X\ =\ f(p_x,\ p_y,\ M )\) is a homogeneous demand function, where \(p_x\)  and \(p_y\) are prices of two commodities X and Y and M is the money income, then prove that the sum of the partial elasticises is equal to zero.
  18. If the demand laws for the two commodities are given by \(q_1\ =\ p_1^{-a_{11}}\ e^{a_{12}\ p_2\ +\ a_1}\) , \(q_2\ =\ p_2^{-a_{22}}e^{a_{21}p_1\ +a_2} \) . Show that the direct price elasticities of demand are independent of the prices, while cross price elasticises of demand are determined in sign by the constants \(a_{12}\) and \(a_{22}\) respectively.
  19. Discuss whether the two goods are substitutes or complementary on the basis of demand function as given below \(x_1 = 6\ e^{-p/100\ q},x_2\ =12 \ e^{-p/100\ q}\) , where p and q denote the prices and \(x_1\) and \(x_2\) are quantities of the two goods respectively.
  20. The demand function for mutton is :\(Q_M = 4850\ -\ 5P_M\ +\ 1.5P_c\ +0.1Y \). Find (i) the income elasticity of demand (ii) the cross price elasticity of demand for mutton at Y (income) = Rs. 1000; \(P_M\) (price of mutton) = Rs. 200; \(P_c\) (price of chicken) = Rs.800.
  21. The demand function for two commodities are given : \(x_1=p_1^{-a_{11}}e^{a_{12}p_2 + a_1}\) and \(x_2 = p_2^{-a_{22}}e^{a_{21}p_1+a_2}\) where \(x_1\) and \(x_2\) are quantities demanded at prices \(p_1\) and \(p_2\) respectively and the rest are constants. Show that (i) the direct partial elasticities of demand are independent of the prices, and (ii) the cross ones are dependent on the price of only one commodity.
  22. The demand functions of two commodities, \(x_1\) and \(x_2\) are \(x_1 = p_1^{-1.4} p_2^{0.6}\) and \(x_2=p_1^{0.5}p_2^{-1.2}\) respectively, where \(x_1\) and \(x_2\) are the quantities and \(p_1\) and \(p_2\) are their prices respectively. Find the four partial elasticities of demand and determine whether the commodities are competitive or complementary.

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