Given chileshe’s utility for ribs , R, and chicken C, is given by U=10R2C, (a) Derive the Marshallian demand equations for R and C. MARSHALLIAN DEMAND FUNCTION is derived by maximising utility, subject to the budget constraint. Our objective function is U= 10R2C Our constraint is M= PR* R+ PC*C We formulate our lagrangian function L as follows: πΏ = 10π 2 πΆ + π(π − ππ π − ππΆ πΆ) The first order condition requires we obtain partial derivatives of the function with respect to QR, Qc and λ and equate to zero. ππΏ ππ ππΏ ππΆ ππΏ ππ = 20π πΆ − πππ =0 ………………EQUATION 1 = 10π 2 − πππΆ =0 …………………..EQUATION 2 = π − ππ π − ππΆ πΆ=0 ………...EQUATION 3 From equation 1 we solve for λ 20π πΆ − πππ =0 20π πΆ = πππ 20π πΆ/ππ = π ……….EQUATION 4 From equation 2 we solve for λ 10π 2 − πππΆ =0 10π 2 = πππΆ 10π 2 /ππΆ = π ……………. EQUATION 5 Now since equation 4 and 5 are both equal to λ, we can equate the two equations as follows: 20π πΆ 10π 2 = ππ ππΆ We then solve for R and C 20RCPC=10R2PR C=10R2PR/20RPC C=R2PR/2RPC C=RPR/2PC…………………………………. EQUATION 6 R2/R= 20CPC/10PR R=2CPC/PR ……………………………………EQUATION 7 To find the demand equation for R, we replace equation 6 into the constraint π = ππ π + ππΆ πΆ π = ππ π + ππΆ ( Rππ ) 2ππΆ The PC will cancel out, so that, M= ππ π + M= ππ π 2 3ππ π 2 2M=3ππ π 2π R= 3π π …………………………………………. DEMAND EQUATION FOR R To find the demand equation for C, we replace equation 7 into the constraint π = ππ ( 2CππΆ ) + ππΆ πΆ PR M= 2PCC+ ππΆ πΆ M= 3ππΆ πΆ π C= 3π ………………………………………………. DEMAND EQUATION FOR C πΆ (b) If income is k90, price of ribs k10, and price of chicken wing is k15, What quantities of x and y will maximise utility? 2π R= 3π π 2∗90 …………………………………………. DEMAND EQUATION FOR R R= 3∗10 = 180 30 = 6 …………………. UTILITY MAXIMISING UNITS OF RIBS π C= 3π ………………………………………………. DEMAND EQUATION FOR C πΆ 90 C= 3∗15 = 90/45=2…… UTILITY MAXIMIZING UNITS OF CHICKEN WE CAN PROVE THAT TOTAL EXPENDITURE IS EQUAL TO INCOME AT THIS POINT. PCC+ PRR=M 15*2 + 10*6=30+ 60=90. (C) what is the level of utility at this optimal level? U=10R2C, R=6, C=2 U= 10*62* 2=10*36*2= 720 (d) Plot the budget line and include an indifference curve on the same diagram to show the optimal combination obtained above To draw the budget line, we need to obtain the two intercepts so that we join then with a straight line. With chicken on the y axis, we know that the maximum amount of chicken wings chileshe can buy will be when all income of 90 is spent on chicken wings of k15 each= M/PC 90/15= 6C With ribs on the x axis, we know that the maximum amount of ribs chileshe can buy will be when all income of 90 is spent on ribs of k10 each=M/PR= 90/10= 9R. Chicken 6 2 U=720 6 9 Ribs (e )Mathematically prove that at the above optimal combination of goods, the slope of the budget line equals the slope of the indifference curve The slope of the budget line can be obtained in two ways. Change in y/change in x=-6/9= 0.67 Or by using the formula –px/py= - Pr/Pc= -10/15= -0.67 Meanwhile, the slope of the indifference curve, MRSxy is given by MUx/MUy, Since ribs is on the x axis, and chicken on the y axis, and since U=10R2C Mux= MU ribs= dU/dR =20RC At equilibrium, R= 6, c=2, Such that MuR= 20*6*2= 240 MuY= MU chicken= dU/dC=10R2 At equilibrium, R= 6, c=2, Such that MuR= 10*62= 10*36=360 MUx/MUy = 240/360= 0.67 HENCE PROVED THAT THE SLOPE OF THE BUDGET LINE IS EQUAL TO THE SLOPE OF THE INDIFFERENCE CURVE AT THE OPTIMALPOINT